The link from the corporate legal department in #5 is a valiant effort, but it's obviously a rearguard action, since the brand name has been used as a generic in normal AE conversation for at least 40 or 50 years, with the consequence that indeed, there is no other word for most people to use to describe the things.
I'm sure their competitors manage to find some other description to put on their packaging, like 'frozen treats,' 'frozen juice bars' (in the case of fruit-juice flavors), etc.
But really, in my experience at least, no Americans have ever used 'ice lolly' (my spell check won't even take it, it suggests 'lollygag') in actual conversation, and it's pretty unlikely that a significant number will ever use 'ice pop.' To me at least, they both just sound British.
So, thank you to AMS in the original post for getting that AE/BE difference right again. Yes, we will still probably call it a popsicle regardless, just as we blow our noses with kleenex. (-: