In my imagination it could have been an undiscovered recording of a mega famous superstar, but I'm not recognizing any pre-fame voice here.
Elaine Baker, who is she? The name is common enough to hit 25.000 at google, and that didn't lead to anything conclusive. There's an opera singer, an actress or radio DJ with the same name here and there, who knows....
The single has no recordcompany label on it. All the label reads is monaural (= mono), the artists name, the composers and some numbers.
The version of Puppet On A String is a bit of an odd blend of romantic Jamaican rhythms and horns with the singer taking a few detours in the original melody and lyrics. Puppet could easily be one of the many versions recorded in Jamaica, but the other track is so un-Jamaican that it makes it a bit doubtful.
Another fact that could point to Jamaica is the misspelling of songwriter Coulter as Coulteer. That occurs on a few other Jamaican coverversions.
The fact that there's no record label is also a point towards Jamaica.
I discarded the possibilty of the record being a cheap pop-tops style copy as the other song is composed by the singer herself and Puppet is far from a plain copy.
A romantic idea that appeals to me is that Elaine Baker just wanted to make a record, spent a few pennies / cents on it and 23 copies were pressed for friends and family. Elvis did it, so why can't Elaine?
Puppet On A String was already featured a zillion times at Eurocovers in 2007. If you use this link you get the posts which feature versions from all over the world.