Differences between $V_{ab}$, $Delta V$, $Delta V_{ab}$, $Delta V_a$ and $V_a$

There isn't a standard, unique notation for this (or technically anything, really). Some of what you have written looks like what I have seen in the past, although...

$V_{ab}$ could instead also be $V_b-V_a$.

With what you have $V_{ab}=-Delta V_{ab}$, which is odd.

I have never seen $Delta V_a$ before, but usually if one is specifying just a single $V$ then they are technically looking at $V-V_0$, where $V_0=0$ is the potential at the reference point where the potential is $0$. This can be at infinity, but it does not have to be (and sometimes it cannot be at infinity).

As for any notation, you should look for where the author(s) of the content define their notation. If it isn't defined, then that's on them, not you. If some text is using all of these notations at once then they are being needlessly complicated. But technically I can't say whether you're right or wrong here; it just depends on how the notation is being used with what you are reading.