Hi
I wanted to let you know what happened. In short: It works!
I redownloaded /moodledata from the old server, transferred the old database to the new server as well as all files.
After I changed the paths in the config everything went smooth. I was able to log in. I put the site in maintenance mode and cleared all caches.
When that was done: Backup database. Then I altered all tables to change from myisam to innodb (ALTER TABLE table_name ENGINE=InnoDB;). Again: Database backup. Repaired the DB, optimized it. Backup DB again.
Then I uploaded the 2.8.1 (new) Moodle into /moodle.new, renamed /moodle to /moodle.old - and /moodle.new to /moodle.
I opened the /moodle and the upgrade started. I needed to update some plugins by hand (remove first, upgrade moodle, install later again).
Result: Everything still smooth! All images are there (as far as I tested). Moodle did not notice any problems with the "new" innodb.
I believe something just went wrong, when I downloaded from the old site. What? I have no idea.
Lessons learned: Keep the old backup until you successfully upgraded.
Thanks again
Olaf