Why do some towers reference departures by compass direction, others by leg of pattern? Got my Private about 6 months ago, and I've always wondered - let's say I request a VFR departure to the South, with the active runway being, say, 2. Some towers will give my departure clearance in reference to the requested compass direction, i.e.
    cleared for takeoff runway 2, right turn to heading 180 approved
whereas other towers will reference it to a pattern leg, such as cleared for takeoff runway 2, right downwind departure approved. Similarly, I've noticed pilots based at airports that use the latter convention will follow that convention when making their departure requests. Is there any rhyme or reason behind which convention a tower follows?
It depends on local custom.

Some airports have multiple runways, so controllers will use compass points instead of pattern legs, and this habit will carry over when they are assigned to a single-runway airport.

Other airports have noise abatement procedures which have specific names. Below are the departures for San Carlos airport, and pilots are expected to use a named departure, such as "Belmon Slough Departure" instead of "right-crosswind departure," and tower will readback the named departure.