I think there are two distinct cross-picking styles. One is an accompaniment style where you play a bass note, then followed by an up-picked chord on an alternate eighth note.
It can be inserted into a standard picking accompaniment for trad or bluegrass. If you employ this with "down-on-a-bass-note and up-on-the-chord", you have a down-up pattern with the pick.

The next "crosspicking" has to do with solo style or fancy runs or fills. Here, the "down-down-up" technique is used to simulate a banjo style roll. A good jazz guitarist can also use an up-up-down technique to run triplets or crosspick the strings. Django used this
on ascendant diminished chords up the neck on three top strings. Sometimes, you glide
the pick over the strings on a "down-down" without using a separate stroke on each "down".

As Rick pointed out, there are many ways to do it.

Frank Hamilton