More than 2 answers to this question.
Snow usually results from cold, dense air flowing under warm moist air, raising it until condensation occurs, as snow. Tops not very high, Warm wet air flowing over cold is the same.
"Thunder snow" results from convective activity that builds towering cumulous. Generally, lightning requires at least 20,000 feet of vertical air flow to develop. This snow may be mixed with sleet or large hailstones. A friend of mine was on a ridge in the Rockies during basic training, they were hit by thundersnow, and three of them were struck by lightning and killed. Naturally, these storms will have the same characteristics as a regular thunderstorm, but the lack of very thick layers of moist air prevents the summertime colossus. A strong wind pushing a thick layer of wet air over mountains would be the typical source of this type of snow.
Others here will doubtless inform us on other sources of snow, and how high they extend.
Edited to add, I have flown in snow at 7,500 feet over Virginia, just west of Richmond, and the reported cloud layer was at 12,000 feet. I had to descend to 3,500 feet to reach visual conditions in rain. No precipitation was in the forecast. ATC treated me very kindly, I was on Flight Following and a VFR flight plan.