I’m trying to find the name of a particular musical technique that you hear in in a lot of bluegrass fiddle music. It’s when, instead of hitting a note (or chord) precisely, you start a little lower and sort of slide up to the note. It’s indicated in the score by a small diagonal line preceding the note. (I don’t mean the gliding from one note to the next, as a pop singer would do.)
Also, is there a different term for starting higher and sliding **down **to the note?
All I’ve ever heard this called (in Irish fiddle music) is a slide. But I’m googling, just to be sure…
Just to clarify, I assume that we’re not talking about either a glissando or portamento, which are full note-to-note slides (unless of course we’re talking about an appoggiatura glissing to the main note).
Which is a longwinded way of saying I don’t know what the right answer is either, but I do know some of the wrong ones. 
You’re after either:
Appoggiatura:
An accented appoggiatura is a grace note which takes the accent and part of the time-value of the following principal note. The appoggiatura slides down to the principal note.
Acciaccatura
A short accented appoggiatura sliding up to the principal note.
Or you could go with the simple grace note.
If that’s not it I’m stuck.
Just to agree with Gyrate, and disagree with Caught@work, if it’s not ‘beginning’ from another note, it isn’t a glissando, portamento, appogiatura, acciacaatura, or grace note. All of these require a beginning and an end - as far as I can see, what is described is an end, with an indefinite start.