At this point, probably your best bet is the extender. If you had planned ahead, you would have removed the existing flange, extended the pipe, tiled around it, then installed a new one. Various things can cause a toilet to leak...no, it should not happen, but occassionally does. If all goes well, yours never will. If the flange is on top of the floor, it is more likely that you'd see the results before it rots out the subfloor (can't tell if you are on a slab or wood). With the flange lower, you need a double or extra deep wax ring to make the seal, and if it does leak, you have a well there where it can accumulate. The thicker the wax ring, the more possibility you could have a blow out if you get too aggressive plunging it if it ever gets clogged. It also means the bolts are longer, which can allow the toilet to move a little (not much, but it may not take much to break the seal).
I'm not trying to scare you...millions of toilets never leak, but it can happen. In your case, now, I'd clean up the existing flange, put a good quantity of silicon there, and screw the extender down into the subflooring. I think the existing ring on that flange is painted aluminum, so it shouldn't rust.