Aaaaaaaaaaaaah!
So the mystery is solved.
So, even by pressing the same button:
the first time I have uploaded the sketch using the USBasp programmer,
the second time I have uploaded the sketch using the AVRISP MKII programmer.
More important: why the first upload stopped the serial line from working?
My sketch started normally, doing his initialization stuff, display init, etc. etc. Then it sent a "wake-up" message to my OBD-device over the serial, but no response.
Maybe some different memory I/O mapping?
Maybe the programmer was assuming that all of the 32K was available and this caused (somehow) the wrong behaviour?