I'm very curious where this came up.  In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives.

Let $a_0,\dots,a_{N-1}$ be the roots of unity that are visited along the knot, in (cyclic) order.  Suppose we have a minimal representative for some non-trivial knot.  Then we cannot have $|a_k - a_{k+1}| = 1$ for any $k$, as otherwise we could replace this pair $a_k, a_{k+1}$ by a single root of unity (for $N-1$), adjusting the other roots of unity as appropriate.  A little more subtly, we cannot have $|a_{k-1} - a_{k+1}| = 1$ either, as then we could again delete $a_k$ from the sequence to get a smaller representation.  With these simple constraints, the smallest possible sequence for a non-trivial knot is the one you found for one of the trefoils with $N=7$.  There are several possibilities for $N=8$, including the one you found for the other trefoil.  I've included a very short Haskell program below that computes this.  The possibilities for $N=8$ are
$$
(2,7,5,3,1,6,4,0)\quad
(2,5,7,3,1,6,4,0)\quad
(3,6,1,4,7,2,5,0)\quad
(2,6,4,1,7,3,5,0)
$$
$$
(3,1,6,4,2,7,5,0)\quad
(2,4,6,1,3,7,5,0)\quad
(3,5,1,7,4,2,6,0)\quad
(4,2,7,5,1,3,6,0)
$$
$$
(3,1,5,7,2,4,6,0)\quad
(5,3,1,6,4,2,7,0)\quad
(2,4,6,1,3,5,7,0)
$$

For the second question, I have never heard of this representation before.

Here is the code, for anyone interested.

     -- A (partial) circular stick representation is a list of integers,
     -- the order of the roots of unity to visit in order
     type CircStick = [Int]
     
     -- The next element ak after a partial representation a1, ..., a{k-1} 
     -- must satisfy
     --   (a) ak has not already been seen
     --   (b) |ak - a{k-1}| > 1
     --   (c) |ak - a{k-2}| > 1
     -- There are a few more "easy" constraint, eg the first and last entries
     -- cannot differ by one.  We do not impose those constraint here.
     nexts :: Int -> CircStick -> [Int]
     nexts n [] = [0]
     nexts n [a1] = filter (\a -> abs (a-a1) > 1) [0..n-1]
     nexts n (a1:a2:as) =
       filter (\a -> not (elem a as)) $
       filter (\a -> abs (a-a1) > 1) $
       filter (\a -> abs (a-a2) > 1) $
         [1..n-1]
         
     completions :: Int -> CircStick -> [CircStick]
     completions n as | length as >= n = [as]
     completions n as =
       concat [completions n (a:as) | a <- nexts n as]
       
     -- Impose final constraints:
     --    (a) Last entry cannot be 1
     --    (b) Take entry that is lexicographically less than its reverse
     --    (c) first and next-to-last entries cannot differ by one
     circSticks :: Int -> [CircStick]
     circSticks n = 
       filter (\as -> abs ((as!!0) - (as!!(n-2))) > 1) $
       filter (\as -> as < tail (reverse as)) $
       filter (\as -> head as /= 1) $
       (completions n [])

**Edit:** For those interested, here are the 108 possibilities for $N=9$.  I hope there's some way of checking what these are more efficiently than just going through them by hand.

[[2,7,5,3,8,1,6,4,0],[2,7,5,3,1,8,6,4,0],[2,5,7,3,1,8,6,4,0],[2,7,5,1,3,8,6,4,0],[2,5,7,1,3,8,6,4,0],[2,6,8,3,5,1,7,4,0],[2,7,5,3,1,6,8,4,0],[2,5,7,3,1,6,8,4,0],[2,7,5,1,3,6,8,4,0],[2,5,7,1,3,6,8,4,0],[3,8,6,1,4,7,2,5,0],[3,6,8,1,4,7,2,5,0],[3,7,1,4,6,8,2,5,0],[2,8,6,4,1,7,3,5,0],[2,6,8,4,1,7,3,5,0],[2,7,4,1,6,8,3,5,0],[2,4,8,6,3,1,7,5,0],[2,4,6,8,3,1,7,5,0],[3,8,1,6,4,2,7,5,0],[3,1,8,6,4,2,7,5,0],[3,6,1,8,4,2,7,5,0],[3,1,6,8,4,2,7,5,0],[2,8,4,6,1,3,7,5,0],[2,4,8,6,1,3,7,5,0],[2,6,4,8,1,3,7,5,0],[2,4,6,8,1,3,7,5,0],[3,7,1,4,6,2,8,5,0],[2,7,4,1,6,3,8,5,0],[3,8,5,1,7,4,2,6,0],[3,7,5,1,8,4,2,6,0],[3,5,7,1,4,8,2,6,0],[4,7,1,3,5,8,2,6,0],[4,1,7,3,5,8,2,6,0],[4,8,2,5,7,1,3,6,0],[4,7,2,5,8,1,3,6,0],[4,2,7,5,1,8,3,6,0],[2,4,7,1,5,8,3,6,0],[2,8,5,3,7,1,4,6,0],[2,7,5,3,8,1,4,6,0],[3,8,1,5,7,2,4,6,0],[3,7,1,5,8,2,4,6,0],[2,7,5,3,1,8,4,6,0],[2,5,7,3,1,8,4,6,0],[3,1,7,5,2,8,4,6,0],[4,2,7,5,3,1,8,6,0],[3,5,1,7,4,2,8,6,0],[4,2,7,5,1,3,8,6,0],[2,4,7,1,5,3,8,6,0],[3,1,5,7,2,4,8,6,0],[5,8,3,1,6,4,2,7,0],[5,3,8,1,6,4,2,7,0],[3,5,8,1,6,4,2,7,0],[5,3,1,8,6,4,2,7,0],[3,5,1,8,6,4,2,7,0],[5,1,3,8,6,4,2,7,0],[5,3,1,6,8,4,2,7,0],[5,1,3,6,8,4,2,7,0],[4,6,1,3,8,5,2,7,0],[4,1,6,3,8,5,2,7,0],[4,6,2,8,5,1,3,7,0],[5,8,2,4,6,1,3,7,0],[5,2,8,4,6,1,3,7,0],[2,6,4,8,1,5,3,7,0],[4,2,6,8,1,5,3,7,0],[2,4,6,8,1,5,3,7,0],[2,6,4,1,8,5,3,7,0],[2,4,6,1,8,5,3,7,0],[2,5,8,3,6,1,4,7,0],[5,3,1,8,6,2,4,7,0],[3,5,1,8,6,2,4,7,0],[5,1,3,8,6,2,4,7,0],[5,3,1,6,8,2,4,7,0],[5,1,3,6,8,2,4,7,0],[4,2,8,6,3,1,5,7,0],[2,4,8,6,3,1,5,7,0],[4,2,6,8,3,1,5,7,0],[2,4,6,8,3,1,5,7,0],[3,6,1,4,8,2,5,7,0],[3,1,6,4,8,2,5,7,0],[2,8,4,6,1,3,5,7,0],[4,2,8,6,1,3,5,7,0],[2,4,8,6,1,3,5,7,0],[2,6,4,8,1,3,5,7,0],[4,2,6,8,1,3,5,7,0],[2,4,6,8,1,3,5,7,0],[2,6,4,1,8,3,5,7,0],[2,4,6,1,8,3,5,7,0],[5,7,3,1,6,4,2,8,0],[4,6,1,3,7,5,2,8,0],[5,3,7,1,4,6,2,8,0],[3,5,7,1,4,6,2,8,0],[6,4,2,7,5,1,3,8,0],[5,7,2,4,6,1,3,8,0],[6,2,4,7,1,5,3,8,0],[2,6,4,1,7,5,3,8,0],[5,2,7,4,1,6,3,8,0],[6,3,1,5,7,2,4,8,0],[6,1,3,5,7,2,4,8,0],[2,7,5,3,1,6,4,8,0],[2,5,7,3,1,6,4,8,0],[3,6,1,4,7,2,5,8,0],[6,2,4,7,1,3,5,8,0],[2,6,4,1,7,3,5,8,0],[4,2,7,5,3,1,6,8,0],[5,3,1,7,4,2,6,8,0],[3,5,1,7,4,2,6,8,0],[4,2,7,5,1,3,6,8,0],[3,1,5,7,2,4,6,8,0]]