There is actually a thread in the Seabourn forum about Seabourn dropping "The Yachts of...".
From my standpoint I think both are distinctions without a difference for the vast majority of people. Most Seabourn reps and agents have always referred to Seabourn as "Seabourn". SeaDream Yacht Club is referred to as "SeaDream" and I don't recall many actually grabbing onto the yacht or club aspects. (Though the ships are clearly yacht-sized...actually being worked on the same yacht yards some of the superyachts I am involved with used.)
Similarly, I do not see many folks referring to Azamara Club as anything other than Azamara .Maybe that will change, but let's face it: Most people will shorten or abbreviate out of convenience.
For me it is what happens onboard the ship that will give a line its cache. I haven't yet experienced Azamara Club to see what the "club" experience is or how it is delivered.