I installed diagnostics from the latest release on both websites:
http://test.lebl.cz/?diagnostics
http://test.theflyingman.cz/?diagnostics
Basically, both servers have
suhosin, which definitely is the issue. How you got one of them working and not the other, I don't know, but they might as well have different suhosin-configurations. Also, strange servers ... They are "similar", but "different" servers setups ... They have different PHP versions, one has Debian (the other likely linux), they have different Apache versions, different PHP memory limits, and one has Mod Headers disabled. To be honest, this doesn't seem to be part of a stable fundament that a professional host would be offering, with mixed, outdated configs that vary across domains.
Is this some private/custom hosting setup? Unless you manage to figure out how you managed to get one login working, which is likely related to suhosin config, then there is little more we can do.
PS! I actually fixed mod_rewrite to work on both websites by adding "RewriteBase /" in the .htaccess file for both websites.