I finally got my website note org-capture template working for new files.

Originally, I only had it working on existing files using this snippet.

("wne"
 "Creates a note in an existing file"
 plain
 (file buffer-file-name)
 (file ,(file-name-concat website-template-dir "note.txt")))

This template uses the file name of the current buffer to select the insertion target.

Getting it to work with a new file specified by the user, requires a small tweak.

("wnn"
 "Creates a note in a new file"
 plain
 (file (lambda () (file-name-concat website-note-dir (format "%s.md" (read-string "Enter file name: ")))))
 (file ,(file-name-concat website-template-dir "note.txt")))

At first, I thought I had to use the function target type since I wanted to use a function to capture the file name. That didn't work.

I then realized, I could still keep the file target type. However, to fill in the file name, I could use a function which takes in user input.

Now that I got this working, I've also done the same for my reponse template.