Hi, Ahava. I'm not exactly sure where some bellydance "experts" get that, other than Bollywood dance has a few hip-driven articulations, but not that many. It's been pointed out in past threads that Asian dances have more hand articulations, while Middle Eastern/North African dances are more hip-driven (and of course, Egypt is in North Africa ). You are exactly right in being able to distinguish the differences.
For whatever reason, some people want to tie India to Middle Eastern dance, where no such connection exists. It's only wishtory on their part. Those who teach this have not done their homework, and refuse to give up their personal notions. Maybe someone else here can better elaborate on why some are so determined to believe that bellydance has any ties to India (and absolutely NO disrespect here on my part against India. I love that country and its dances, too---I just don't try to fit a square peg into a round hole).
Although I can't explain why some are set on believing what they do, I have read some of their "theories." One author of a book I own states that in the 5th century A.D., the Romany from India travelled through Turkey and Persia, then on to Egypt and North Africa, and from those places, spread bellydance everywhere they travelled. No disrespect to the Romany, but, yeah, no. -_- The author cites no resources for what she wrote, just stating, "many people agree."
While many cultures share similar articulations in dance, each culture puts its own spin on these movements that are peculiar to their country. Just because two or more cultures share a similar movement vocabulary doesn't mean they ran over to each other and said, "Heeeeeey! Lookie what WE can do, let's ALL do it!"
Aaaaaaaand with that, I think I'll get off my iPad and get some sleep, I think I'm losing where I was going with this and am beginning to not make sense to myself! @____@