An Android ROM is basically the Android operating system you are running on your device at the moment. HTC uses Sense UI, while Samsung uses TouchWiz, but of course not on all of their ROMs since there are Developer and Google Edition devices. I know this is probably difficult, but would you somehow be able to record your gameplay and show us the lag you're talking about? And one more thing, did you check for any new updates using the Software Update option under Settings > About? I am not going to lie to you, I did not keep the stock 4.1.2 ROM long enough to test it out after it settles for days, in fact I installed a custom recovery and a custom ROM the same day I got the device, but about two weeks later I went back to stock in order to try and I found that it is fast and stable, but not as much as 4.2.2 based ROMs.
Have a look here:
[GUIDE] Complete Flashing Guide | Rooting | Going Back To Stock | 05.05.2013 - xda-developers
You should start reading about rooting, recoveries and installing custom ROMs - 4.2.2 ones will definitely speed up your device a whole lot, however you'd have to go back to stock if you want to take advantage of the warranty. In my opinion and since you're still stock, I'd pay your carrier a visit and show them the lag if it's truly noticeable, and then show them a video of a lag-free HTC gameplay on YouTube. Here is one:
HTC One - Gaming Performance - Review - YouTube and he plays Temple Run as well, among other games of course.