Well, you can look up the specs on your current servo and see if you can find one thats a couple of mm shorter than the one you were trying to use or you can buy a low profile servo like THIS Savox one.
First thing I would do is try a new receiver. If that works then you are correct, bad RX. If that doesn't work then Id start testing with known good servos.
Quick question but have you checked to make sure your batteries have a good charge on them?
I went and looked at them because I have both sets of connectors on hand and I believe you're correct. There are some slight variations but they are 95% the same. It should work.
JST connectors are electrical connectors manufactured to the design standards originally developed by J.S.T. Mfg. Co. (Japan Solderless Terminal). JST manufactures numerous series (families) and pitches (pin-to-pin distance) of connectors.
JST connectors are used in many types of products, and...
You'll need to either crimp/solder on a Futaba connector. Or you can take the wires out of the red housing and put heat shrink around them and then just plug the individual wires in to the RX.
Theres no need for the 3rd wire or an adapter, its just a signal wire. Like WF said, make sure you have positive hooked up to the middle pin and make sure the negative wire is hooked up to the negative pin which is the pin toward the outside of the receiver housing.