A quick update:
I used a new brush this morning, and the lather was already better quality. At least I could still see the lather on my face after shaving one cheek :)
I took a serious step back and only shaved the whiskers of the right cheek using the straight, as adviced in the beginners guide of the forum. I used a somewhat larger angle for shaving (up till this morning, the angle was zero, so the razor was held parallel to the face, with the spine touching the skin). I also went across the grain the second time with the safety razor instead of against the grain. No cuts and no redness of the face. Steady she goes ...
Jan