Oh well, I feel lazy but I try to give my 2cents, even this propably would need much longer answer.
with 1d, fan and sun is the same move. (at least as I see this) On 2d word sun is used for patterns based on two handedfour beat siteswap pattern 312s2s .
Where sun happens on 22 part and 31 is the set up.
And just to be clear how these sun/fan terms are used at the moment with 2d.
sun = 312s2s, 4 beat pattern
fan = 2s2s, 2 beat pattern
31 is the two handed siteswap notation for basic shuffle.
2rbw, 2 or 2rfw, 2 would be the hover speed up,
continuous speed pulls would be 2rbw/rfw231
22 would be the minicolums with 2 diabolo
rbw= right back wrap
rfw= right front wrap
fan is just continuous 22 movement. diabolos stay on same place on the string but string rotates a circle.
nice patterns from here are doing regular 22 (minicolums) and going in to different versions of fan, (right out, right in, left in, left out) keeping the pattern as 2222s2s as a five beat version and 22s2s as much more difficult 3 beat version. some 31222 versions are fun too, 3122s2s is really good fun as the sun part changes direction so to speak is antifan or antisun. (s after siteswap, explains the sun/fan)
At this point I point out my point here: sun and fan are for me same things. they describe same movement, it is just difference between the siteswap of the pattern. I know I have said this before but I don't really enjoy naming the tricks. As usually the trick names don't explain anything about the movement/trick itself.
2handed notation for 3low would be 51? or (4x,2x) depending how you like to think of it, or do you prefer sync or async.
cascade is 333
with columns you have lot of ways to go. 333 is possible as 4440, 423, they are different but just coulumns don't give enough info.
box is (4, 2x)(2x,4)
Hover is multiplex pattern. [2rbw, 1] 3,
other nice hover patterns with 3 are [2rbw, 2],2 which is minicolums with hover, and [2rbw, 3],1 which is hover with anti shuffle.

for 3d suns/fans with 2handed siteswap notation, here's some variations without feeding.
regular 3d sun/fan would be [2s,2s],2s or 2s,[2s,2s]