With the red laser I suspect it’s thermal expansion of the cavity which changes the resonant frequency. The LED is a broadband emitter but constructive interference occurs only when the cavity is an integral number of wavelengths. Change the length and the resonant wavelength changes.
The Nd transition is at a frequency governed by an electronic transition involving a 4f orbital. Those orbitals are largely isolated from their chemical and thermal surroundings so the transition is very much narrow band which doesn’t change greatly with temperature. The pumping diode is a broadband source but that doesn’t matter as long as it emits strongly enough at the laser frequency.
In a previous life I used to play with a tunable laser — a dye laser pumped by a fixed frequency argon-ion laser in that instance. It was tuned by changing the length of the cavity.