For once the orange livebox triumphs with connecting the ps3
BT and 02 are the same company and both use bt phone lines so not really suprising these are the two most people have problems with
the reason i think people are getting confused is because o2 bought the mobile side of bt which was bt cellnet who also had the online genie sim cards. i can't remember whether bt & o2 are still linked some how but i don't think they are. i think it was a sale rather than just a company name change.
as for isp's most of them use the bt phone lines, (obviously the cable users don't though) but their are plenty that do use the bt phone line. i have been disconnected a few times on GTA 4 but i don't know why really as i have played GT5P online loads & been disconnected less times than i have on GTA 4 which i have only been on about 8 or 9 death matches, 1 team death match & once on the mafia mission thingy. it could just be server related or isp. at weekends or certain times of the day (usually evenings) i think certain isp's cap your speed if your a heavy user.