As you've discovered, there is no standard meaning to each contact in a TRRS connection. Some devices are built to be backwards-compatible with older TRS connections, and as such have ground in position 3, putting the mic in the sleeve. Other devices put ground at the sleeve and the mic in position 3. (And if your device also outputs video over TRRS, all bets are off.)
The cable you linked doesn't specify its internal wiring. It is possible that your cable's TRRS sleeve is ground, but your phone is providing ground on the second ring. You can use a multimeter's continuity checker to see which TRRS contact is connected to each contact on the TRS end.
But for this particular use case, I think a standard TRS-TRS cable should work just fine. I don't know of any TRRS audio I/O device that can't handle a TRS connection shorting the microphone to ground, as this is a fairly common use case. If you're getting bad sound quality, it likely won't be because of mismatched connections, and probably has a different root cause. Do other TS/TRS microphones, plugged into the recording device, record with acceptable quality? If not, the problem is likely inside the recorder. Can your audio output device plug into a standard set of computer speakers and produce good audio? If not, the problem is likely inside the phone. If both work, try a higher-quality cable.
I don't know which recorder you have, but it's also possible it only has a TS connection jack, and though it's probably just leaving your R channel floating, it's theoretically possible that your recorder is shorting the phone's R output to ground. If it's a TS jack, you'll probably want a stereo-to-mono converter to make sure you capture both channels regardless.
