Here's the deal Wannabe...

You WILL have to report that you did not complete two training events and were basically forced to resign in both instances; whether or not that shows up on PRIA, the next airline WILL find it, because they are required by law to get your training records from any and all previous 135/121 and sometimes even 91 operators that you have flown for, depending on who's hiring you.

The correct course of action, without ANY question is to fully disclose that you did not complete training. How you want to spin that is your business, but you MUST disclose it if you don't want to be (further) blackballed by being fired for lying; which you most certainly will be by any 135 or 121 operation that requires a copy of your training records.

Aviation is a very unforgiving business in many aspects, and the best thing you can do for yourself (aside from the obvious) is to operate in complete transparency with your next employer (or PSA if you go back there). Will 2 incomplete training events be a nail in your coffin professionally? Hard to say; depends on the employer. If you do well going forward, that will help to mitigate the impact of those unsat events. If you don't, I agree with others that you should probably do something else.

I can tell you with certainty that being terminated for lying WILL be the final nail in the coffin; so don't do anything that could be construed as such, okay?

I hope that this helps to clarify things for you. I wish you the very best.