The Rift(s)?

[SilverArrow20XX]

This got me thinking, so I went and scrubbed through some videos and found the following:
When locations in FF14 are introduced, the format is that they have the name of the specific place in big letters on the bottom, and then above it in smaller letters they have the name of the broader place name that the specific place is a part of. So in the image, we see that the specific place Old Sharlayan exists in the broader place The Northern Empty. Thus when the Interdimensional Rift is introduced, these titles tell us that the Interdimensional Rift is simply one place in the Aetherial Rift. So that's probably a big part of what's been confusing, is that when people say "rift" it can refer to different things. But at least this in-game introduction clarifies that while related, these are distinct concepts.

The Interdimensional Rift transcends space/time. All universes exist within it, but it can also be anywhere you open a door to it.
Rather than thinking of the Aetherial Rift as a bigger place that the Interdimensional Rift is inside of, it's more like it's just where Omega created his pocket dimension, with "Interdimensional Rift" being a generic term for any interdimensional space of a less specific "location".
Using an Ocean metaphor, I think the Aetherial Rift is the shallows where you can still touch the bottom, and so has a permanence of structure shaped by the nearby Shards.
As 3 dimensional beings, we can't perceive the entirety of the Rift and need pathways to see the connections. To go "deeper" you need to move through higher dimensions.

When he deleted his pocket space, we are still in the Rift (but in the deep), and Alpha is able to summon us back to an area where there is space. The Aetherial Rift.
Omega's portal to the Rift is still open, so Hraesvelgr is able to simply fly into the hole and get us.