Script algorithm that modifies the input layer and returns it as its output.

model_with_rename_field_script.model3

<!DOCTYPE model>
<Option type="Map">
  <Option name="children" type="Map">
    <Option name="qgis:basicstatisticsforfields_1" type="Map">
      <Option name="active" type="bool" value="true"/>
      <Option name="alg_config"/>
      <Option name="alg_id" type="QString" value="qgis:basicstatisticsforfields"/>
      <Option name="component_description" type="QString" value="Basic statistics for fields"/>
      <Option name="component_pos_x" type="double" value="741"/>
      <Option name="component_pos_y" type="double" value="494"/>
      <Option name="dependencies"/>
      <Option name="id" type="QString" value="qgis:basicstatisticsforfields_1"/>
      <Option name="outputs" type="Map">
        <Option name="results" type="Map">
          <Option name="child_id" type="QString" value="qgis:basicstatisticsforfields_1"/>
          <Option name="component_description" type="QString" value="results"/>
          <Option name="component_pos_x" type="double" value="745"/>
          <Option name="component_pos_y" type="double" value="602"/>
          <Option name="default_value" type="invalid"/>
          <Option name="mandatory" type="bool" value="false"/>
          <Option name="name" type="QString" value="results"/>
          <Option name="output_name" type="QString" value="OUTPUT_HTML_FILE"/>
        </Option>
      </Option>
      <Option name="outputs_collapsed" type="bool" value="true"/>
      <Option name="parameters_collapsed" type="bool" value="true"/>
      <Option name="params" type="Map">
        <Option name="FIELD_NAME" type="List">
          <Option type="Map">
            <Option name="parameter_name" type="QString" value="newfieldname"/>
            <Option name="source" type="int" value="0"/>
          </Option>
        </Option>
        <Option name="INPUT_LAYER" type="List">
          <Option type="Map">
            <Option name="child_id" type="QString" value="script:rename_field_1"/>
            <Option name="output_name" type="QString" value="OUTPUT"/>
            <Option name="source" type="int" value="1"/>
          </Option>
        </Option>
      </Option>
    </Option>
    <Option name="script:rename_field_1" type="Map">
      <Option name="active" type="bool" value="true"/>
      <Option name="alg_config"/>
      <Option name="alg_id" type="QString" value="script:rename_field"/>
      <Option name="component_description" type="QString" value="Rename Field"/>
      <Option name="component_pos_x" type="double" value="441"/>
      <Option name="component_pos_y" type="double" value="408"/>
      <Option name="dependencies"/>
      <Option name="id" type="QString" value="script:rename_field_1"/>
      <Option name="outputs"/>
      <Option name="outputs_collapsed" type="bool" value="true"/>
      <Option name="parameters_collapsed" type="bool" value="true"/>
      <Option name="params" type="Map">
        <Option name="INPUT" type="List">
          <Option type="Map">
            <Option name="parameter_name" type="QString" value="inputlayer"/>
            <Option name="source" type="int" value="0"/>
          </Option>
        </Option>
        <Option name="INPUT_FIELD" type="List">
          <Option type="Map">
            <Option name="parameter_name" type="QString" value="inputfield"/>
            <Option name="source" type="int" value="0"/>
          </Option>
        </Option>
        <Option name="NEW_FIELD_NAME" type="List">
          <Option type="Map">
            <Option name="parameter_name" type="QString" value="newfieldname"/>
            <Option name="source" type="int" value="0"/>
          </Option>
        </Option>
      </Option>
    </Option>
  </Option>
  <Option name="help"/>
  <Option name="modelVariables"/>
  <Option name="model_group" type="QString" value=""/>
  <Option name="model_name" type="QString" value="model_with_rename_field_script"/>
  <Option name="parameterDefinitions" type="Map">
    <Option name="inputfield" type="Map">
      <Option name="allow_multiple" type="bool" value="false"/>
      <Option name="data_type" type="int" value="-1"/>
      <Option name="default" type="invalid"/>
      <Option name="description" type="QString" value="input_field"/>
      <Option name="flags" type="int" value="0"/>
      <Option name="metadata"/>
      <Option name="name" type="QString" value="inputfield"/>
      <Option name="parameter_type" type="QString" value="field"/>
      <Option name="parent_layer" type="QString" value="inputlayer"/>
    </Option>
    <Option name="inputlayer" type="Map">
      <Option name="data_types" type="List">
        <Option type="int" value="5"/>
      </Option>
      <Option name="default" type="invalid"/>
      <Option name="description" type="QString" value="input_layer"/>
      <Option name="flags" type="int" value="0"/>
      <Option name="metadata"/>
      <Option name="name" type="QString" value="inputlayer"/>
      <Option name="parameter_type" type="QString" value="vector"/>
    </Option>
    <Option name="newfieldname" type="Map">
      <Option name="default" type="QString" value=""/>
      <Option name="description" type="QString" value="new_field_name"/>
      <Option name="flags" type="int" value="0"/>
      <Option name="metadata"/>
      <Option name="multiline" type="bool" value="false"/>
      <Option name="name" type="QString" value="newfieldname"/>
      <Option name="parameter_type" type="QString" value="string"/>
    </Option>
    <Option name="qgis:basicstatisticsforfields_1:results" type="Map">
      <Option name="create_by_default" type="bool" value="true"/>
      <Option name="default" type="invalid"/>
      <Option name="description" type="QString" value="results"/>
      <Option name="file_filter" type="QString" value="HTML files (*.html)"/>
      <Option name="flags" type="int" value="8"/>
      <Option name="metadata"/>
      <Option name="name" type="QString" value="qgis:basicstatisticsforfields_1:results"/>
      <Option name="parameter_type" type="QString" value="fileDestination"/>
      <Option name="supports_non_file_outputs" type="bool" value="true"/>
    </Option>
  </Option>
  <Option name="parameters" type="Map">
    <Option name="inputfield" type="Map">
      <Option name="component_description" type="QString" value="inputfield"/>
      <Option name="component_pos_x" type="double" value="172"/>
      <Option name="component_pos_y" type="double" value="261"/>
      <Option name="name" type="QString" value="inputfield"/>
    </Option>
    <Option name="inputlayer" type="Map">
      <Option name="component_description" type="QString" value="inputlayer"/>
      <Option name="component_pos_x" type="double" value="143"/>
      <Option name="component_pos_y" type="double" value="380"/>
      <Option name="name" type="QString" value="inputlayer"/>
    </Option>
    <Option name="newfieldname" type="Map">
      <Option name="component_description" type="QString" value="newfieldname"/>
      <Option name="component_pos_x" type="double" value="606"/>
      <Option name="component_pos_y" type="double" value="302"/>
      <Option name="name" type="QString" value="newfieldname"/>
    </Option>
  </Option>
</Option>

rename_field.py

# -*- coding: utf-8 -*-

"""
***************************************************************************
*                                                                         *
*   This program is free software; you can redistribute it and/or modify  *
*   it under the terms of the GNU General Public License as published by  *
*   the Free Software Foundation; either version 2 of the License, or     *
*   (at your option) any later version.                                   *
*                                                                         *
***************************************************************************
"""

from qgis.PyQt.QtCore import QCoreApplication
from qgis.core import (QgsProcessing,
                       QgsProcessingAlgorithm,
                       QgsProcessingParameterVectorLayer,
                       QgsProcessingParameterString,
                       QgsProcessingOutputVectorLayer)
from qgis import processing


class ExampleProcessingAlgorithm(QgsProcessingAlgorithm):

    INPUT = 'INPUT'
    INPUT_FIELD = 'INPUT_FIELD'
    NEW_FIELD_NAME = 'NEW_FIELD_NAME'
    OUTPUT = 'OUTPUT'

    def tr(self, string):
        """
        Returns a translatable string with the self.tr() function.
        """
        return QCoreApplication.translate('Processing', string)

    def createInstance(self):
        return ExampleProcessingAlgorithm()

    def name(self):
        return 'rename_field'

    def displayName(self):
        return self.tr('Rename Field')

    def group(self):
        return self.tr('Example scripts')

    def groupId(self):
        return 'examplescripts'

    def shortHelpString(self):
        return self.tr("Rename Field")

    def initAlgorithm(self, config=None):
        """
        Here we define the inputs and output of the algorithm, along
        with some other properties.
        """
        self.addParameter(
            QgsProcessingParameterVectorLayer(
                self.INPUT,
                self.tr('Input layer'),
                [QgsProcessing.TypeVector]
            )
        )
        self.addParameter(
            QgsProcessingParameterString(
                self.INPUT_FIELD,
                self.tr('Input field name')
            )
        )   
        self.addParameter(
            QgsProcessingParameterString(
                self.NEW_FIELD_NAME,
                self.tr('New field name')
            )
        )   
        self.addOutput(
            QgsProcessingOutputVectorLayer(
                self.OUTPUT,
                self.tr('Output layer')
            )
        )

    def processAlgorithm(self, parameters, context, feedback):
        source = self.parameterAsVectorLayer(
            parameters,
            self.INPUT,
            context
        )
        input_field = self.parameterAsString(
                parameters,
                self.INPUT_FIELD,
                context)
        new_name = self.parameterAsString(
                parameters,
                self.NEW_FIELD_NAME,
                context)
        
        # Rename the field
        source.startEditing()
        source.renameAttribute(source.fields().indexFromName(input_field), new_name)
        source.commitChanges()

        return {self.OUTPUT: source}