Yes, more precisely the town needs to have a medical spa ("Heilbad" in German) [1].

Which can be a medicinal spring, but also other potentially beneficial effects [2], there are towns on the sea, which have the prefix "Seeheilbad" [3], which is purely based on the positive climatic effect of the nearby sea.

And then, this is Germany after all, there is a long list of other boxes that need to be ticked, see [4].

[1] https://de.wikipedia.org/wiki/Bad_(Kurort)

[2] https://de.wikipedia.org/wiki/Heilbad

[3] https://de.wikipedia.org/wiki/Heiligendamm

[4] https://de.wikipedia.org/wiki/Seeheilbad

Yes, that's indeed true. The English city of Bath, home of the famous Roman baths, is also known as "Bath Spa". Glad we've cleared this up :|

Not a German speaker, but is baden the plural form then? I feel sometimes spa towns have baden in the name and others are just "bad."

The plural would be "Bäder", but I'm not aware of any town using that plural form. It would be very strange, too, because a town is a single entity.

Baden-Baden is the only one I know that comes close, and https://en.wikipedia.org/wiki/Baden-Baden#Name explains how it got that name. It's not the spa prefix, but part of the name proper.

Also, "Baden" is a region in the south-western state of Baden-Württemberg that probably has lent its name to other towns.

It is home to one of the two big dialect families in Baden-Würrtemberg: Badisch, the other being Schwäbisch/Swabian.

Well there's also Wiesbaden in Germany, which is pretty well known. There are some others as well closer to the Czech republic.

From your link:

"In modern German, Baden is a noun meaning "bathing"[4] but Baden, the original name of the town, derives from an earlier plural form of Bad ("bath")"