Second, have you tried to delete the cache (.cache/com.ubuntu.telegram)? That will erase your secret chats too, so you can make a copy before deleting it.
Curious as to why your getting a different result though.
EDIT: I'd also be curious as to your opinion of Touch in general?
My opinion: I love it. I'm not a huge fan of Unity, or Ubuntu, and it's definitely in the beta stage. I like that most of the problems that I have were solvable on my own via shell-script or whatever hacks I come up with.
In that thread someone said deleting .config/com.ubuntu.telegram/<number>/auth.lock solved the problem with Telegram not starting [1].
So look for .lock file/s and delete it/them.
I still love Debian way more than Ubuntu, but I think they're doing a great work with Touch.
[0] https://bugs.launchpad.net/ubuntu/+source/qtbase-opensource-...
[1] https://bugs.launchpad.net/ubuntu/+source/qtbase-opensource-...
Once in a while I got a blank screen opening Telegram, but closing it and open it again solves the problem.
Are you running OTA-15?