Hey there fr3nd
Good catch. I had forgotten about がる. If you build them both up, 欲しい + がる + いる and 欲しがる + いる in the correct forms, you get the exact same kana: 欲しがっていました. With that in mind it comes down to meaning, and, well, they pretty much mean the same thing too. I wouldn't be surprised if the origin of the verb 欲しがる is the combination of the adj 欲しい and the verb がる. Which leaves us with both being valid, and so...
I was mistaken, the site is correct with the word it has linked. Thank you for pointing that out.
As a side note; this is one of the reasons I wanted to see all the auxiliary verbs linked. It can be hard to spot them like this. That said, I'm getting better at it and don't really need them linked anymore. This will probably be the last one I get wrong.