I've been going round in circles with this, and I'm not sure I've got to the bottom of it, but what keeps coming up is the amount of misinformation contained in web resources, which are all I've really got to go on.
For example, if you look for the lyrics to Arthur Gunter's Baby, Let's Play House, you will usually see the line about the pink Cadillac, which was not in the original song, but was a later Presley modification. The best way to find the correct words is to actually listen to the track.
As for Eddy Arnold, you will find sites which say that he also recorded Baby, Let's Play House, which as far as I know he didn't. But as mentioned above, in 1951 he did record a completely different song, written by Cy Coben, called I Wanna Play House With You. These two songs seem to get regularly confused with each other.
I suppose you can understand why. For one thing the words to Gunter's Baby, Let's Play House do not contain that exact phrase at all, but do contain the line "I wanna play house with you", which is the title of the Eddy Arnold song.
It gets worse if you look up the lyrics to Eddy Arnold's I Wanna Play House With You. Some sites, whether or not they get the title right, list the words to Baby, Let's Play House instead, invariably in Elvis's "pink Cadillac" version.
So while I may be missing something, is this the key to the puzzle, in that eight of the artists have recorded songs or albums with a reference to a pink Cadillac in the title or words, while two, Arthur Gunter with Baby, Let's Play House and Eddy Arnold with I Wanna Play House With You, have recorded songs which (at least by some people) are thought to contain mention of a pink Cadillac, but in reality they don't?