If I could have one bike that was good for everything I'd wanted to use it for, I'd own two
No really, I'd prefer to have to deal with only one bike but it is nice to have a spare when one is temporarily out of commission for whatever reason.
In practice it's a little different. I used to have one fair weather road bike and one winter/off road bike but on a whim I put together a fixie out of the crashed remains of another bike. Now I really like riding that bike but don't want to give up either of the other two. So now I have 3 bikes.
It's a sickness that many of us share called N+1.
Personally I don't think you
need a separate bike just for cold and rainy weather. I never hesitated to take my "fair weather" bike out in the rain and have ridden it in temps well below freezing as long as the roads were clear. You might get some enjoyment out of having a 2nd bike but that's a different question.
If you bought your Trek recently and haven't done much cycling before that you might want to hold off on another purchase for a bit. The reason I say that is that after riding your Trek for awhile you might get an itch to try something a little different. So rather than just getting a beater you could get a 2nd bike that provides you with a refreshing change of pace.