Testing 3D content is the way to go. If it works with that, I don't think there's any other reason to replace.
I had a couple old monoprice HDMI cables (6-7 years old I'd guess) that ended up causing 3D not to work (Bluray player couldn't detect the 3D TV) when I got my new TV that I had to swap out. Was glad to be rid of them anyway since they were stiff, braided cables that were a hassle to get in place and keep out of view anyway due to being so stiff.
How can I tell if the HDMI Cable I already own is "high Speed" or not? Are there markings on the cable that could be interpreted?
Use the cable. There is a 99% chance it will just work. If it does not work, you will know immediately because you will see it not working.
There is no difference in HDMI cables. If you can see the picture without visible dropouts or sparklies, the cable is working at 100%. No other cable will display a better version of that picture. You're simply wrong if you think there is a better digital cable than one that is already working. (Oh, and plasma didn't die because of logistics problems, nor does OLED ship in big boxes because it comes from Korea.)