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You’re spot on. The flats I occasionally swap on and off the bike have a very different stack height than the SPDs/shoes combo I use. If for more than a ride to the park, I need to adjust saddle height. The fore-aft has never been an issue for me, but height for sure. Cleat position, combined height of pedal/cleat/shoe, type of flat pedals, thickness of shoe sole, etc., all contribute.
I tend to position myself based on what ‘feels right’ for pedaling. Seems to work OK for me.
Opened a can of worms here. Seems there's a few variables: different pedals probably have different stack height differences on either side, shoe stack height differences, cleat position, rider flexibility/ sensitivity to changes/ pedalling style, etc.
Mainly I commute on flats. I did big miles at the weekend on clipless which I can confirm is a bad idea. Moving saddle further back and up helped. Probably an element of fit being wrong in the first place.