I don't know if I take the Red Cross job as fact. It's far too colorful (all that red, you know ;))) and there is no corroboration - that I am aware of - aside from Bukowski's own stories.
In the first ("on being 20") he says he was 20 years old, and left from the (Temple Street?) rooming house (after a visit from his mother, which is also telling). In the second ("the troops"), he says he went after WWII broke out, which would make him at least 21, and possibly 22, if he left later in the year, after August 16.