Does the name of the rift have any influence on its content?

Solution 1:

Official information on this is pretty hard to come by, but there are 2 seperate pieces of information that point towards Nephalem Rifts being completely random and having content independent of their name. The first one is from the BlizzCon 2013 RoS Panel:

So what we have done is we have taken all of the tilesets, all of the dungeons in the game, and we organize them in new and interesting ways. So we have taken the cathedral, all the exteriors and all the monsters and shuffled them up.

So every time you do one of these, it’s random. [...] So every time you transition from one level to another, it can be a different dungeon and it will have a different combination of monsters and we mix up the type of dungeons you are going through.

For example we’ll start with the cathedral, we can go into a root cave after that, go to crypt, or you can go to just one level which is just a big exterior zone. Or you can go from exterior into a Zoltun Kulle dungeon or something like that. So there are all kinds of different ways we put this together and every time you do them it is different.

In the Beta forum there has also been a post by Tsamis that heavily points towards the fact that Rift names have no meaning at all:

Going forward, please try to include a link to a screenshot showing the environment of the Rift you were in when you encountered the bug. If you are unable to provide a screenshot, please do your best to describe what the environment looks like (i.e. "The area looks similar to the Keep Depths"). The names of the Rifts are randomized so including the name listed in the upper right corner of the screen does not really help us when we are trying to reproduce these issues internally.