Paint the door frame white, remove light pink brush strokes

Yes, there was a botcher at work. That's just not acceptable – you also have your pride when you do something yourself.

I can also imagine that the dispersion paint comes off with warm water. I would try rubbing thoroughly with warm water and a cloth to see what holds... maybe you'll get lucky.
I didn't sand my door frames (back then) beforehand. I always use lye to clean the surface and for door frames I would then prime.

Then apply a matte varnish.
I would personally pull straight down at the edge with a flat brush. If you also have a steady hand, it will work. If not, then tape off with painter's tape and remove it while still wet. You get evenness with a foam roller. First paint the inner edges with a round brush and then apply the paint to the adjoining surfaces with the foam roller. And always with the grain, meaning up and down, not sideways.

I think that once the paint smudges are wiped away, the frames will be white enough again.

Here are a few pictures. I think I still have quite a bit ahead of me. With a brush, I made the paint a bit wet (lukewarm). After that, I waited for half an hour. I started to rub with a cloth. No chance. I took some universal oil, sprayed it on, and used a very small pointed round wedge to scrape off some of the paint. Unfortunately, the steel rust is very visible. Can it be made invisible afterwards by painting or something similar?

I have found a solution on how to easily remove the pink stains. My mother gave me the advice. On Friday I will share it here.

May I guess, please...???
A magic eraser!

and took a very small, pointed round chisel, with which I scraped off some of the paint. Unfortunately, the steel rust is very visible.

Oops. That was a bit brutal o.O
You completely chipped off the paint instead of trying to rub it off... It will only become really invisible with quite some effort, if at all. Unfortunately, I have few tips except sanding – but that usually causes further chipping next to it...

I'm curious about your mother's trick! Good luck!