Thanks! I first updated to 1.7.1 before doing anything.
Then checked that these look as they should, which they seem to do:
[attachment 530 Skrmavbild2012-02-26kl.18.37.58.png]
Then I deleted all piwik_archive_* tables.
Then I set Piwik settings Allow Piwik archiving to trigger when reports are viewed from the browser to Yes.
Then I clicked on the Piwik logo to go to the front page. I reloaded the database tables and saw four new piwik_archive_* tables, only tables for *_2012_01 and *_2012_02, nothing for 2011.
After this I ran
/usr/bin/php5 www//misc/cron/archive.php --url=http://
in the SSH terminal, but this changed nothing.
The end result: The changes for site #1 to site #2 got through and all visits that was erroneously collected/reported is now visible for the right site. However, one problem still exists: Logs from before january are missing.