Funny... I woke up and started trying to fix the Steam.
I did some changes, compiled it, test it - and whoa! It worked!
Ok, let's check what part of my change really fixed it.
So I remove some part of it, compiled it, test it - ok, still works, so it was the second thing I changed.
Just to be sure, let's revert it all back to the original and test it. Hm... it still works.
Probably I copied the wrong file or compiled file stayed the same.
I clean and recompile it all, copy again. And -- it still works. Wut? Even with original file from Plugin Updater it works.
Hm, so perhaps Steam servers got update and fixed their own issue. Ok, good.
... then I looked into the SVN log and saw unsane's commit from the night with message "Steam: reworked login".
Anyway, thanks unsane for fixing it