Reservation "under the name", "in the name", or "by the name" of Ms. X

Solution 1:

For AE (midwest, Calif.), I recognize all three as acceptable, but would tend to use "under". It's what I encounter most frequently, but I'm also imagining the host looking up the reservation name in a book with a bunch of tabs with letters on them (even though that's not actually what's happening most of the time). You can also just say "I have a reservation under Franklin".

If I were actually showing up at a restaurant, I'd most frequently use "for" -- "Hi, we have a reservation for Franklin."

"In the name of" has a slight implication that you're making the reservation with a name other than your own.

Solution 2:

I do think there is a difference between 'in the name' and 'under the name'. For instance, the secretary calls the hotel and asks to make a reservation in the name of her boss Mr.Cullen. On the other hand, let's imagine I go to the conference and I make my reservation or get registered under the name of my company MTC. That's how I see it.