The reason I have pretty well determined the bad transistor is a PNP is because of the plus side of the bad electrolytic is connected to the collector.
If the + of an electrolytic is connected to the Collector of the transistor, then the transistor is NPN.
You have another way to find out as well, since you stated you have a working identical relay. Open that one up and use the ohmmeter on Diodes scale to find out if the transistor is NPN or PNP. The majority of single transistor relay coil drivers encountered are going to be NPN type, BTW.
You mention the ubiquitous 2N2222. I would find that one to be rather light as a relay driver, suggest you look at the 2N4401 instead. Or any of a number of NPN amplifying or switching transistors in that class, including EIAJ parts such as 2SC828, 2SC945 - watch for pinout differences and install accordingly.
If you aren't sure how to tell an NPN bipolar from a PNP using your ohmmeter's diode scale, now's the time to get over that simple hurdle. The first thing to know is which lead of the ohmmeter actually has the Positive voltage on it, with any of the modern DMM's it is very likely to be the Red probe. With Positive going lead on the Base of the NPN transistor,you should read about 0.6 to 0.7V from Base-to-Emitter and Base-to-Collector. Of course, the opposite would apply to the PNP type.