I don't get the impression that the name of the product is up for debate, only how to "market" it, and that's obviously fine.
However, "Media" is very vague, and will draw LOTS of results from any number of intended searchers. "Center" is even worse. "Media Center" is nearly as vague. It could mean a physical location where media is stored (a warehouse), a physical place where the news media comes together, the name of a competing product, and more.
there is little that you can do to eliminate these other results no matter what *else* you add to that part of the name.
there is nothing current (that I'm aware of) that jriver (or JRiver) could be confused with, except maybe the iRiver player, which I think is no longer available/marketed.
Think about Xerox. its' not ever going to be confused with anything else, it's unique, like JRiver
perhaps "media center" is not helping the cause at all. yes, it helps describe the product, which is usually a good thing, but it also describes your competitors products, and generically describes any media player, and the things above.
I suggest the official product name be shortened to JRiver
that's it. that is the name of this product. in a few years all relevant internet references to this product will only reference this new name, and you will grow out of the confusion that "media center" has surrounded you with.
"What {media/music/video/whatever} player do you use?"
"JRiver, it's great, you should try it."
a search for JRiver would certainly never be done; who adds a period to a search? I don't.
j river would not be an uncommon search, but it's 2 pages before anything other than this software comes up and "D&J River Ranch" is not likely to take any traffic from you.
leaving "media center" in the official name will ALWAYS take traffic from you and/or add confusion to people.
the folks over at foobar don't have this issue, it's unique, therefore stands alone/apart from anything else. EASY to find.
My Google results for j river media center are "About 10,400,000 results"
My Google results for j river are "About 88,300,000 results"
My Google results for foobar are "About 4,970,000 results"
My Google results for jriver are "About 1,390,000 results"
Just my $.02