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While I'm satisfied (despite the fringe) with shoots from a partially visible moon with a cheap 500 mm lens:

1/100 ISO 100 f/?

I cannot understand why the image quality is extremely diminished when doing the same with an almost fully visible moon:

1/80 ISO 100 f/?

I suppose that fully visible is much brighter and when I try to compensate the situation by changing the aperture these effect show up.

Is there an obvious explanation or do I just mess up the focus every time?

EDIT:
I repeated the process with an APS-C instead of full-format camera, now one of the images looks better (showing more details):

1/250 ISO 100 (APS-C)

I'm not sure whether the crop factor allows me to focus more accurate or this is due to the 50% magnification. At least it shows more details.

What makes the difference on partially and fully visible moon?

In a word: shadows.

I cannot understand why the IQ is extremely diminished when doing the same with an almost fully visible moon.

The second image does appear to suffer from lower sharpness and overall quality. However, even if the technical image quality factors were equal, most importantly, a full moon will appear flat and uninteresting, as compared to a gibbous (i.e., 3/4-ish) moon.

We see a full moon that is lit directly "overhead". The mountain and crater rims do not cast any shadows that give texture and depth to the moon's surface. Thus, we only get tonal information from the albedo (reflectivity) of the local regolith in parts of the image.

However, with a partial moon, the surface is more side-lit, thus casting shadows. These shadows provide vital depth clues to our eyes, and greater tonal variations. Even if the technical image quality is the same (i.e., same accurate focusing, correct exposure, no motion blur, etc.), a partial moon's greater tonal and texture variations will make for a more apparently higher-quality moon image.

Your images provide for excellent comparison. Looking at Tycho Crater (the large impact crater in the south-southeast view, with large whitish ejecta streaks emanating from it): Notice in the gibbous phase image, Tycho crater has a distinct rim and bowl, and is surrounded by lots of smaller impact craters. Whereas, in the full moon image, the shape of Tycho crater is apparent, but it has no depth, no sense of being an obvious bowl. The smaller impact craters immediately surrounding it are nearly invisible, and the general area surrounding Tycho is just a smudge of middle gray.

The third image has substantially higher quality than the second image. However, even with its increased clarity and dynamic range, notice that the area immediately surrounding Tycho crater still doesn't exhibit much depth. The crater rim has more definition than the second image, but to my eye, doesn't exhibit nearly the same dimensionality and character as the first image does.

The full moon is illuminated by the sun which is directly overhead, i.e. noon. Mountains on the moon's surface, at noon local time, cast no shadows. When the moon appears to be partially illuminated, gibbous, 1st.or last quarter, 1/2 moon, crescent etc. This local time is such that the sun casts shadows. A view of the moon from earth is enhanced by shadows as they heighten the illusion of depth.