OK, so here's an update... In short - I still don't know whether or not my cars are OBD2. Read on for more detail...
I drove the 99GLS to my friend's place and we tried his three OBD2 scanners. None of them would work, although he wasn't sure if that was simply because he had them all configured for his Toyotas. I had assumed that they would at least generically identify the presence of an ECU if the car was OBD2... Anyway, after fiddling around with things for quite a while and essentially concluding that the car mustn't be OBD2, he found a freeby car scanner App in the "play store" on his Android phone, so with nothing to lose he downloaded it and behold! - it worked!! See photo attached for the App details. Basically, he just scrolled through the vehicle configuration options in the App and picked the '99 Sonata 2.5l GLS option (the oldest one listed), and it then read the codes, identifying the bank1 O2 sensor and the temperature sensor with errors (which is very consistent with some of the erratic engine behaviour). It even gave some links to click on taking us to photos of the sensors concerned in-situ. It did however only seem to work by Bluetoothing with one of his three readers - the OBDLinkMX+ (photo of the packet attached). He said from memory, it cost him about $45, although didn't recall whether than was USD or AUD. At that price, I didn't care - it was worth that just to get one that I knew would at the very least work with the free scanner app.
With both us having run-out of time, he loaned me the OBDLinkMX+ so I could download the App on my own phone and spend a little more time playing with it before deciding to order my own. However, unfortunately, I've not since repeated that initial success we had. I've also not been able to find that particular scanner as cheaply as he recalls...