OK! I have this problem too, with many mods. For some reason, I'll click on internet games, and the connection thing will never complete. The log in will never be successful. So I press cancel and it takes me straight to the server list. When I filter a game, any game, and update, it gives me a list of counter-strike servers. Very annoying.
I have no idea WHY this happens, but here's what I do: I have a shortcut for counter-strike on my desktop and a shortcut for Day of Defeat. I'll click the counter-strike shortcut and try to get in. If it fails and I don't log in after I click internet games, I quit counter-strike, and then open the DoD shortcut. If clicking on internet games doesn't work there, I quit DoD and open counter-strike again. Most of the time I only have to go Open CS--->quit---> open DoD before it works. I dunno if it has to do with switching mods or just quitting half-life and starting it up again. Either way it works for me.