whether the websites was actually the clients to take
Yes, it absolutely was.
You were paid to design the site (which you did) and then had an additional deal to host the site.
If the client chooses to terminate the hosting that’s entirely reasonable.
Let’s say you’re a struggling artist…I pay you to paint my portrait…you do and I pay you for it…it’s still my picture.
I might even let you hang it in your gallery (because I don’t have anywhere to hang my picture at the time)…but time moves on and I get a bigger house with plenty of empty wallspace…..I come to you and ask for my picture to hang on my wall….do you feel taken advantage of now?
No…of course not….
Same principle applies here. Move on.