It does not have to be written down to be specified or to be specific.

I suppose not. But if it isn't, the controller must explain what it means
each time he uses the phrase.

No, just each time he uses the phrase with a pilot that doesn't
understand it. If most pilots flying there are based there, and the
phrase is used often enough (and consistantly enough) and CFI/radio
osmosis occurs (this is not at all farfetched), then a lot of radio
time is saved, because most of the time it will not need to be explained.

I do agree that this is not sufficient justification to invent phrases
willy nilly. But it's really no different from "N42312 report the tanks".

When the controller said "modified straight in", and no other
instructions, I assumed that to mean that I could fly at pilot's
discretion. For those of you with more hours than I (300), please
correct me if I'm wrong.

I would disagree (which is different from "you're wrong" . I would
take "modified straight in" to be a specific and common course which
makes sense given the geography of the place, and if I didn't know
what course that was, I'd ask. Absent altitude restrictions, I'd
assume altitude is at the pilot's discretion (within the FARs).

Jose
--
Freedom. It seemed like a good idea at the time.
for Email, make the obvious change in the address.