No. This definition is clearly wrong.
Ok, the subsequent logic is good enough but the premise is false. That’s what is wrong, the premise.
So the definition does not apply to the past.
The past is clearly over and done with so whether finite or infinite it would have to have had an end. At least now it has to have an end.
So the past does not go on at all. It’s finished. Whether finite or infinite, the past is finished. It does not go on.
On the other hand (or end?), counting an infinite past would have a beginning but no end. God could count it, if He could be bothered, but not us. But we’re very clever and we have an infinite amount of time in front of us to find a smart way maybe to count the past.
A possible definition of the past may be time that just ended, so the past definitely has to have an end and this applies to both a finite and an infinite past. A finite past would have a beginning to, while an infinite past wouldn’t.
So that’s it. And pretty much everyone here should be able to agree with that.
EB