"If there's two labels, the lower one is from a PSP1000 and the higher one is from a PSP2000, because the 2000 is quite a bit more battery efficient"

The only one I can comment on is the battery mart one - they asked for more than the battery for shipping. I think I paid over $60 CAD at the end of import duties and taxes. this was double-recommended about five years ago from social media and a forum I regularly visited. I kept it fully charged using an external PSP battery charger that was fed off an official PSP power source.
Almost all of the others I possessed had been run dry at least once due to neglect. At least 1/3 were mine from new, and I regularly cycled them, tho I can't say they never ran low.