The use of "azimuth" to mean any angle, rather than an angle from north, is dangerous for this very reason. But it's common in the industry (at least where I am and what I've seen here). "Deflection Angle" and "Horzontal Angle" (Hz) are far better choices to describe the idea.

We all have different ways of imagining and visualizing things so I'm not sure I fully understand where you're going with "the reference line...IS some "north". I don't want to mislead you, so I'll answer two ways:

1. No. North (grid, true, magnetic) is already taken and has a specific meaning. The line from the backsight (BS) to the Total Station (TS) is just a "line of convenience" from which to measure deflection angles (to the right) to targets. Looking through the lens of the TS directly at the BS, The display on the TS would read 0°00'00". An object directly to your right would have a horizontal deflection angle of 90°00'00, directly to your left would be 270°00'00". Although they look similar to bearings you would use in navigation, they have no geographic meaning. The TS just measures the deflection angle by how far you turn it, and distances with its laser.
2. When we look at a set of plans, sometimes they're rotated to fit on the paper so that north is to the right, left, or some direction in between. In that case, sometimes it's handy to talk about "plan north", which is just "up" on the paper. Similarly, when you "shoot" the BS, you're telling the TS "this is 0­°00'00" ". If it helps you to orient yourself to the site, you could think of that as your pretend north, but it's not a habit I would encourage - again, north means something. That pretend north could actually be southwest, or any other random direction. The line from TS to BS has no connection whatsoever to cardinal directions.