I can't duplicate that behavior here*, but I have heard of it.. Reports of Region editing causing odd behavior like Regions moving unintentionally go back through several versions of LPX, and I've not heard of a definitive remedy, though that doesn't mean someone hasn't found oneâ€”you may want to search around a bit, you might find some useful information..
* I guess this could be a system-specific or procedure-specific issueâ€”I tried cutting two or more Regions with Snap on & off (I remember some saying that Snap affects the behavior); with anchors intact & changed (I could still see the small anchor dots in the Regions); the cuts always were where I made them and the Regions did not move or shift unintentionallyâ€”FWIW, this was with Logic 10.4.6 in Mojave 10.14.5 in a fairly fresh Template (it's always helpful to include what version(s) you're working in).
I'd suggest maybe to try troubleshooting by starting with a simple recording (one track), make some cuts to test; add additional tracks but no anchor adjustments and make cuts; if no bug, try Snap on & off and different Snap settings; if still no bug, try adjusting the anchors and make cuts again, etc.. At some point assumedly the bug will rear its head, and maybe you'll be able to better isolate any specific contributing factors/settings..
A couple of general Logic observations that may or may not be related.. Make sure that the Project starts at Bar 1 and not beforeâ€”earlier start points are known to cause odd behaviors; See if having the Regions start right on downbeats or barlines has any effect; Since you didn't mention it I assume you had Flex Time offâ€”that's also reported to cause similar issues when editing..
Good luck with this, and please, if you are able to find a solution, post and let us know.