Indeed, all running and 1.3.

Now, that's upstairs on the desktop, you've said it's OK to install Steam and access the game from my laptop downstairs too.  Given how long it takes to download LH again, can I somehow copy what I've already got installed from upstairs, then perhaps install Steam client properly on top, and it would accept that it doesn't need to re-download the whole game?  That would save a lot of time...  Thanks.

https://support.steampowered.com/kb_article.php?ref=8963-EIKC-3767

Here is a faq that will answer quite a few of the question you may have lingering. If you log into steam and a second user logs into steam, it disconnects the first user and vice versa. They suspect they do not do what you suggest because of the large number of problems that arise with general internet connections throughout. The faq directly addresses what you say harpo99999.

Personally, the thing just pops up and says the account is already in use or some such and prevents log-in. I haven't been locked out of steam for it though.

edit: I haven't tried copying the file data stuff over from one computer to another, but I suspect there is other files not in the game directory that steam checks for and so you might have to re-download the entire game, unless you find an acceptable way around it in the steam forums.