I use Steam from about 5 computers in different physical locations, and it never asks me to log out the way you described. I just log in on the machine I'm currently using, that's all.
Oh, ok - this has changed since I last tried it. I still don't think it's brilliant that you can't have your software open on more than one machine (several years ago you could at least have Steam open on multiple computers if they had the same external IP).