Floor plans of a single-family house on 640m² with low eaves height

0.7 * 640 does result in the above number, doesn't it?

Oh, somehow I mixed up every possible value in the opening...

I didn’t. You just multiplied the floor area ratio of 0.7 by the wrong value. The basis is the floor area of the ground floor or basement (depending on whether with or without cellar). And with a floor space index of 0.4, that is a maximum of 256 m² (640 m² x 0.4). 256 m² x 0.7 = 179.2 m² would be the usable area on the upper floor with a maximally large basement/ground floor.

For me, with "exploitation of the floor area ratio," it was not about a maximally large upper floor related to the plot, but about a maximally large upper floor related to the basement/ground floor. "Thanks to" the maximum eaves height and a maximum roof pitch of 30°, I thought I could already calculate how high my knee wall could be with a rectangular floor plan. No chance - I can’t manage it.

Um, I think you are mixing something up there.

Floor area ratio = floor area ratio = maximum buildable area of the plot, including terrace etc.
Plot ratio = plot ratio = maximum gross floor area that can be built on the plot, where only full floors count.

Wikipedia can explain this to you in a bit more detail.

maximum large upper floor (OG) relative to the basement/ground floor (UG/EG)

That is then more the question of whether two-story construction is allowed - then the upper floor area (OG) can equal the ground floor area (EG) - or not. In your case, the upper floor (OG) may not be a full story, meaning it may only have 2/3 of the area of the full story beneath it (here EG), whereby this definition varies from federal state to federal state.

Apparently, I really haven't quite figured it out yet. I'll follow up!

It's not a big deal, but that's why I was a bit confused that you want to use the [Geschossflächenzahl] (see above). Usually, it's the [Grundflächenzahl] that's the problem (with smaller plots of land, probably not in your case), not the [Geschossflächenzahl].

Maybe not bad, but somehow embarrassing for me. I value content that is factually correct. If you haven't understood things, you should clarify that.