I'm not asking you who's on second
Q. Is it third base line or third baseline? For example: Our seats were on the third base line. – from Des Moines, Iowa on Thu, Jun 11, 2015
A. It's baseline in the "baseball" entry, so make it third baseline.
It may be "baseline" in the baseball entry, but it's "third-base line" in real life for the same reason a businessman who runs a small business is a "small-business man" (I'm looking at the 1986 edition under "s"). The compound can't do its magic if it isn't a compound. You're not even being inconsistent -- this is what compounds do to reduce ambiguity. (A book written for schoolchildren in the primary grades is written for ...)
If your seats are behind the third-base dugout (not the third basedugout), you're along the baseline that goes from home to third, which is the first baseline if you're going left to right but the second baseline if you're counting right to left. So who decided to go with "third baseline"? I don't know.