Diet is the oldest of these words and originally the broadest. It meant a whole way of living, of which food was one element alongside sleep, exercise, air and work. A writer prescribing a diet in that older sense was describing a routine.
The narrowing of the word to mean food alone, and then to mean a restricted intake in particular, is a comparatively recent event, and the older meaning survives in phrases like a varied diet.
Regimen and the medical tradition
Regimen kept the broad sense longer. It belonged to a medical tradition in which the ordering of daily life was the treatment, and its texts read like timetables. Food appears in them as one column among several.
The word survives mainly as a formal alternative to routine, which is a considerable demotion for a term that once organised an entire branch of practical medicine.
Reducing, slimming and the commercial vocabulary
Reducing and slimming arrive with commerce. Both are twentieth-century terms, both are transitive in feeling, and both were used heavily in advertising before they were used much anywhere else. They describe an operation performed on a body rather than a way of living.
The two words also date a text almost instantly. A paragraph using either one can usually be placed within a couple of decades on the strength of the word alone.
Why the older words are still useful
Recovering the older senses is not nostalgia. The broad meaning of diet described something real: eating sits inside sleep, work and movement, and a vocabulary that separates them entirely has to keep reinventing ways to put them back together.
Watching a word narrow is one of the more reliable ways to watch an idea narrow, and this particular word narrowed quickly and in public.