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I have figured out how to save only the features of a vector layer that intersect a rectangle (I'm doing something similar to the example below):

TransformContext = QgsProject.instance().transformContext()
SaveOptions = QgsVectorFileWriter.SaveVectorOptions()
SaveOptions.driverName = "GPKG"
listOfLayers : list[QgsMapLayer] = layerIdsToLayers.values()

for layer in listOfLayers:
  currentExtentsToSave = QgsRectangle(0, 0, 1, 1)
  SaveOptions.filterExtent = currentExtentsToSave
  SaveOptions.onlySelectedFeatures = False

  SaveOptions.actionOnExistingFile = QgsVectorFileWriter.ActionOnExistingFile.CreateOrOverwriteLayer
  fileName = "Example.gpkg"
  QgsVectorFileWriter.writeAsVectorFormatV3(layer, fileName, TransformContext, SaveOptions)

With some testing, I confirmed that the calculation to determine overlap checks if any parts of the geometry of each feature overlap with the extents provided in the SaveOptions.

How can I modify the SaveOptions so that the calculation performed to determine an overlap only uses the center of each feature's geometry?

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    Unfortunatelly, I have not yet found a way to check geometry topology between a point and a QgsRectangle in the filterExtent. However, one could try the QgsFeatureRequest with .setFilterExpression() together with the materialize to create a temp layer. Additionally, it is better to switch to writeAsVectorFormatV3 as v1 and v2 are deprecated since QGIS 3.40 and QGIS 3.20, correspondingly. Commented Apr 15 at 7:01

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A solution I found with the help of @Taras is to use the "Extract by expression" processing function:

To use it, I first added some variables to the layer that I wanted to extract features from:

upperLat = 32.494
lowerLat = 32.3
upperLong = -117
lowerLong = -120

layer = QgsProject.instance().mapLayersByShortName("MyLayer")[0]
QgsExpressionContextUtils.setLayerVariable(layer, "upperLat", upperLat)
QgsExpressionContextUtils.setLayerVariable(layer, "lowerLat", lowerLat)
QgsExpressionContextUtils.setLayerVariable(layer, "upperLong", upperLong)
QgsExpressionContextUtils.setLayerVariable(layer, "lowerLong", lowerLong)

I then used those variables in the expression:

expression = "with_variable('xCoor', x(centroid(@geometry)),@xCoor >= @lowerLong and @xCoor <= @upperLong) andwith_variable('yCoor', y(centroid(@geometry)),@yCoor >= @lowerLat and @yCoor <= @upperLat)"

I then used the expression in the processing plugin:

processing.run("native:extractbyexpression", {'INPUT':'/home/user/Downloads/layer.000|layername=MyLayer|geometrytype=Polygon|uniqueGeometryType=yes','EXPRESSION':"with_variable('xCoor', x(centroid(@geometry)),\n@xCoor >= @lowerLong and @xCoor <= @upperLong) and\nwith_variable('yCoor', y(centroid(@geometry)),\n@yCoor >= @lowerLat and @yCoor <= @upperLat)",'OUTPUT':'TEMPORARY_OUTPUT'})
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