Another commenter suggested Tiny Core Linux and DSL2024, which are indeed as light as it gets, but you might find yourself limited in what you can do with them, and it's not necessary for those specs.
The next step up would be Q4OS Trinity and antiX. You should be able to get the Spotify app and your preferred web server running on either of those.
I tried this yesterday and it looks like a great alternative to Puppy Linux, or even its base distro AntiX. The software selection is pretty well thought out; I'd never heard of BadWolf but it sounds like an excellent project. Way heavier than the original DSL, but once it's stable it'll be an easy addition to my list of recommendations for really old PCs.
Though - to make a minor nitpick - I have to disagree with the games selection. I can think of plenty of lightweight X11 games in the Debian repo that I'd rather have than volleyball or TuxPuck. (XBill and Koules for example - lots more action in those.)