Well, there are really only two things I care about. The first is that the name of the tartan is preserved. If I decide to create a "Morris of Cowley" tartan, for example, I want it to remain just that. I don't want to find it being sold everywhere as an "Austin of Longbridge" tartan. The second is that I am credited as the designer. I care less about the second than the first, however.

Note: after looking over the Scotweb website some more, it now seems to me that the statements there are mostly Scotweb's way of protecting themselves from copyright claims. That is, Scotweb doesn't assume you hold the copyright on the tartan you supposedly designed in their Tartan Designer: you have to prove that yourself. I am still puzzled by this statement, however:

Note that until a design is recorded as a formal design by being physically produced, any other user can freely copy and commission its production, thereby becoming its legal ‘owner’.
It seems to be at odds with my understanding of copyright law.