Above 6,000 meters (that's nearly 20,000 feet) you're mountaineering, not trekking.

"High altitude" depending on whom you ask is generally taken to mean anything over 2,000-3,000 meters.

there is no clear definition, but for most mountaineering starts when you need technical equipment (crampons, ice axe etc). I do hikes in alps in non-winter almost every weekend in 2000m-3000m band and its pure hiking.

There are peaks higher than 6000m where you will not need this (ie in andes, tibet), and there are much lower peaks where you can't do anything without it.

One example I did - you can do Kilimanjaro without touching any snow or ice, via standard route, and that's 5895m high.

I've always gone with the concept that mountaineering starts where a rope is needed to prevent serious injury in case of a fall. Altitude or even protection is outside of the equation. A sidewalk, if beside a 100-foot drop, counts as mountaineering because the rope is needed.

Nepal's most popular trekking trails, which are below 5600m, are passable even for grandmas.