I know that I actually do anchor in a slightly different spot on my face with an index vs a thumb trigger / handheld because my peep height is not interchangable between the 2 releases. There is a slight variation and I can't swap back and forth. I would have to go and get the 2 styles and really feel where I have my hand in relation to my jawline but there has to be certain "landmarks" that fit the contours of my hand more naturally and also allow me to keep the string in a direct line with my hand and eye rather than out to the side on the most lateral portion of my jaw.
You'll know you are not anchored in a good spot if you feel like you have to tilt your head to the direction of your release hand to get your eye centered in your peep.
Some guys go forward of this point with a short anchor, some guys go behind but they both accomplish the same thing of getting your peep properly aligned with your eye laterally.

As for how your POI is affected with respect to release changes that's going to vary based upon 2 things, location of anchor on your face obviously and release style. The string or string loop doesn't behave the same out of every caliper release. I could take my Scott "Rhino" and my Scott "Wildcat" set for the same barrel length and trigger psoition coupled with the same anchor and the wildcat will shoot an 1" or so left of where the Rhino will just because of how the string loop clears the jaws.

There is no given with respect to swapping back and forth among any releases and POI.........some might find the POI is the same and others completely different. Shooting styles, anchor points, face shapes, jaw styles make that impossible to say that it should or shouldn't.

What was the original question? LOL I guess I should go back and read huh? I got caught up on this page first.