〜ている can indicate a completed-action state, not just in-progress actions.
- 結婚している → is (currently) married
- 開いているお店 → a store that is open
- 太っている → is fat
To disambiguate these states from in-progress, you can use
〜つつある for "happening right now". I've mentioned this in another thread, but don't remember which one at the momemt (will update later if I find it).
- 店が開きつつあります → The store is opening right now (you can see the metal shutters going up, etc.)
But back to the original question, both
〜た can be used for descriptions. It just may seem a little strange to an English speaker at first.
- ワイシャツを着ている人 → A person wearing (in-progress) a white-collar shirt.
- ワイシャツを着た人 → A person who "wore"/put-on (and still has on) a white-collar shirt.
- 太っている猫 → A cat who is currently fat.
- 太った猫 → A cat who got/become (and still is) fat.
AFAIK, the two are pretty interchangeable for descriptions. I'm not sure if either has any limitations of the other. And as I said, it may seem strange at first, but eventually it will become second-nature.