So, I have this Philips LED lamp that was working fine, and then it stopped working. I thought it died and was about to toss it when I decided to try it out on the other socket (two-lamp fixture). It lit up! I put it back where it was when it went out, it didn't light up. So I tried a standard 26w PL-C where the LED was and it worked fine! So what could be the problem? LED lamp or the fluorescent (electronic) ballast?

I understood the post is about a PL-C retrofit LEDs...

Are you sure these LEDs are rated to be used directly instead of the fluorescent?
Because many times they need the fixture rewired to have the 230V (mains) directly connected to the pins.
But if the combination of the ballast and LED is able to somehow give a stable operating point, it may work even with the ballast.

What is likely happening the LED parameters are just on the border of what the fluorescent ballast is accepting as a good vs failed lamp. Normally, with the real rated fluorescent, there is good margin so any small parameter shift is tolerated, but that is apparently not the case with the LEDs.
Initially it was by chance on the "good side" of the cliff and it worked, but by agin some parameters have shifted a bit to the "wrong side", the EOL detection comparator mechanism in the ballast had flipped and shut the ballast down "thinking" the lamp had failed.
As each lamp is different a bit, as well as each ballast (voltages, currents, detection threshold), if the setup is marginal from the start, it may behave like you describe even when technically there is nothing bad with either of the components.

Or do you mean the A60 retrofit lamps?, sounds like some sort of loose connection inside the LED, as just moving it causes it to work or not work?
Does the twin light fixture operate the lamp horizontally, and the single vertical?