Breite = Länge (z. B. Schrank) = length
Breite = Tiefe (z. B. Schrank) = width
Höhe = height

depth = Tiefe (z. B. See), Höhe (einer Schicht)

I disagree. "Depth" is also used for "Tiefe" meaning the distance from the front to the back. You also talk of "the depth of a rack". In the logistics industry, for instance, pallets are stored double-deep in a rack. We use the abbreviation "W x H x D", and this has not led to any problems whatsoever.

lxwxh = length x width x height
lxwxd = lenght x width x depth

Depth or height depends on what you are measuring. e.g. the dimensions of a swimming pool would be given in lxwxd, because the act of measuring starts on ground level and goes down, a building would then be given in height for the same reason, but it goes up.