website updating & updating website

What's the difference between both? My guess is the former one is the action itself and the latter one is focused on the content of the website. Right?

The first is a good noun phrase, Qut, and makes independent sense:

Website updating is an important responsibility of the site administrator.

The second is not complete, and needs to be surrounded with grammar to be useable:

Updating a website is time-consuming. (nonfinite clause as subject)
She is updating her website now.
(part of continuous verb phrase)
This is now an updated website.
(past participle as part of noun phrase)

As you can see, your two forms are not particularly related, grammatically.