Do I have to prove my flight training time abroad when applying for a standalone FAA license?
You can find the detailed requirements for a FAA PPL in Part 61 Subpart E.

The total time of 40 hours can be met by any flight time you have logged, either DUAL or PIC.  (In theory, SIC should qualify too, but I can't think of any legal way for a student pilot to have that.)

DUAL time can only be logged when you are receiving "training" from a CFI, who must endorse each such log entry along with a description of said training (maneuvers, procedures, etc.) in the remarks.  This endorsement is what allows those hours to count toward the 20 hours of "flight training" required for your PPL.

The "solo" time of 10 hours can only be met by flight time where you are the only person in the plane, meaning that it can only be logged as PIC time. PIC time does not need to be endorsed--ever.

Where I think you're getting hung up is that a student pilot cannot legally act as PIC without getting an endorsement before the flight, and those endorsements have limitations.  For instance, my first endorsement only allowed me to fly around the pattern at my home airport, the second allowed me to fly between my home airport and a neighboring airport, and the third was for a specific cross-country trip that I'd previously flown with my CFI.  However, the resulting PIC time was not endorsed by my CFI after those dozen or so flights.

In your specific (and unusual) case, you were able to legally meet the solo time and cross-country time requirements without an endorsement before those flights.  Still, like me, you don't need any endorsements after the flights.  Just logging the PIC time yourself was enough for it to count.

Your records and history will undoubtedly look strange to your CFI (and eventually your DPE), so it's best that you make it easy for them to see which specific log entries you're using to meet each requirement.  I'd suggest marking those entries with sticky flags, noting which FAR(s) they fulfill, and including a translation of the remarks if they're not in English.  Don't note or translate your entire logbook, though; they don't care about anything beyond seeing that you've met the minimum requirements.  And, while it shouldn't matter according to a literal reading of the regs, common sense suggests a preference for time logged in US aircraft (N reg) at US airports (Kxxx).

Finally, FAR 61.103(j) says that any applicant for a US PPL must "Hold a U.S. student pilot certificate, sport pilot certificate, or recreational pilot certificate." There's nothing in the regs that says a validated foreign PPL qualifies, so you'll need to discuss with your local FSDO; if they say it does, it's best to get that in writing so there's no paperwork problems with the DPE on the day of your checkride.