OK, seems like this has to be done as root.

I booted from internal flash and started Xterm.

First I typed: sudo gainroot.
Then: fsck /dev/mmcblk0p2

I get:
/dev/mmcblk0p2 contains a filesystem with errors, check forced.
Pass 1: Checking inodes, blocks and sizes.
Inode 15348, i_blocks is 16, should be 8. Fix<y>?

And now, because I actually don't understand what I am doing: to fix or not to fix?

EDIT: I selected "no" and went through the whole fsck procedure and got about 20 different errors (I answered "no" to all fix-questions). Other problems are of following types (there are a couple of cases of each of these errors, these are just examples):

Deleted inode 9422 has zero dtime.
Entry '32wd_to' in /var/lib/dsme/stats (94404) has deleted/unused inode
Unattached inode 94462
Inode 94540 ref count is 3, should be 2.
Block bitmap differences: -8505 -30734.......
Free blocks count wrong for group #6
Inode bitmap differences: -495...
Directories count wrong for group #0
Free inodes count wrong for group #6
Free inodes count wrong

Which of these errors might be relevant for the booting problem?

EDIT2:
I got adventurous and went through the fsck-procedure again. This time I answered "Yes" to all the fix and remove questions. Now booting from mmc works again!

Thank you Bundyo!