Do 'stay on the list', 'make the list' and 'make the cut' all have the same meaning?

Solution 1:

"Make the list" and "make the cut" are interchangeable. "Stay on the list" probably derives from "makes the list" but it's not something I hear falling into the groove of a familiar expression. Sounds like someone avoided a cliché here.

Bonus extra: And for phrases of opposite meaning, a person who is "on the bubble" is in danger of getting cut.

Edit: "Stay on the list" implies that you've "made the cut/made the list" but are "on the bubble." :=)

Solution 2:

The idea here is there were bars to be scaled in order to get invited to the presidential dinner party. Hence, it was difficult to make the list. Indeed, there is an actual list of guests probably managed by White House staff.

We also talk about making the cut, because guest lists usually begin with many names but the number has to go down to something manageable, hence the cut.

Stay on the list and survive on the list could make sense in this contest, but they are really not idiomatic, i.e. naturally used. In addition, they do not evoke the brutal nature of the intrigues behind these powerful dinner lists.