Are they saying that the clocks in Androids still use quartz, but iPhones use MEMS clocks, even though they both use general MEMS silicon?

Correct. MEMS technology is used in the accelerometers and gyros in the inertial measurement units (IMUs) that are in pretty much every smartphone. Apple decided to switch to using MEMS clocks, probably because it means that they can reduce part count slightly as it would mean that they can incorporate them on the same chip as the CPU or the like.