It's fortunately entirely gone by now. I can't even pinpoint when it went away. I stopped the cortisone spray after the 3rd day, because I really felt like it made everything worse. Time between coughs became longer and longer until it just went away on it's own.Yeah, cough can be an indication of a genuine problem or just an indication of some low-end irritation your body is overreacting to. This is particularly common with lingering coughs.
I'll now wait until November or early December and then get my 4th vaccine shot. Assuming that the negative test on August 2nd marks my actual recovery that should be long enough waiting period.
Today EU approved a new variant vaccine vs. BA.1 that should also be better against BA.4 and BA.5 (I guess the same as mentioned above that was approved by UK earlier). However a "true" vaccine for BA.5 is also already in certification and expected the be approved during September. So I hope I can get that for my 4th shot.