Conversion of a 2013 world crashes as of 1.13

Today I tried to convert a really old map to 1.13 to fix it and play it (it's a pvp game). And the game crashed after ending the conversion. (corresponding files start with first_attempt)

When I tried to load the world again the world loaded, showed up for a split second, then game crashed again. With a different error. (corresponding files start with second_attempt) And the launcher crashed at the same time, too.

I tried converting a fresh copy of the map to 1.12.2 and it worked fine.

I attached the game logs, the crash reports and the map causing the issue.