For classifying peaks height criteria can be helpful. However, meaningful height criteria vary considerably from one range of mountains to another. Height criteria for one range may not be applicable to another due to differences in the height of the surrounding landscape.
For example, in the UK a threshold of 500 metres makes sense for a meaningful peak. On the Isle of Mull, Ben More rises 966 metres more or less from sea level. In Bavaria, the Voralpen rise up from the Bavarian plateau, starting at about a height of 500-700 metres. A height criterion of 500 metres would be ridiculous there. However, the Voralpen are roughly a similar size to some of the Scottish Highland peaks, but starting from a higher initial level, so a height criterion of more than 1 000 metres would make sense.
Taking a concrete example, Ben More on the Isle of Mull rises 966 m directly from sea level. In Bavaria, the Wallberg in the Voralpen rises above Tegernsee (height 726 m) to a height of 1 722 m. The height difference between the Wallberg and Tegernsee is 994 m. Thus Ben More and the Wallberg have a similar size in relation to their environments but are in quite different height ranges.
Similarly, the main ridges of the Alps rise from surrounding countryside having an elevation of 200-1000 metres. In these ranges, the interesting peaks have elevations above 2000 metres. In Switzerland, France and Italy there are a number of summits with over 4000 metres in height. Therefore height criteria of 2000, 3000 and 4000 metres would be interesting.
The Himalayas are bounded to the North by the Tibetan Plateau which averages a height of 4500 metres and valleys in Nepal that are over 1000 metres. However, the interesting summits are clearly much higher than in the Alps. There are the famous 8000er summits but they are few in number. A more interesting height criterion would be 7000 m or 6000 m.
While prominence criteria can be readily used consistently across different mountain ranges using the 1-2-5 principle, it is clear that height criteria need to be specific to a particular mountain range or geography.