I think this is a way to explain how it works in detail (based on existing rulings and what I found to be logical).
Hopefully this helps.
Here is the key distinction:
A) "current site" refers to where the site marker is (i.e. the site the fellowship is "closest to" during the movement process). Therefore, this always applies and controls which site's game text is active.
B) "at" refers to the current site when the fellowship is not in the process of moving to or from it. Therefore, anything that checks where the fellowship or a minion is "at" does not apply while the fellowship is moving.
<Fellowship or Regroup phase begins>
...
...
Free Peoples player announces they are moving.
------------------------The fellowship is "at" the old site until this point.----------------------
(So anything that checks the site the fellowship is currently "at" does not apply after this point.)
1) A Shadow player places the next site card, if needed.
2) Perform "When you move from..." actions.
3) Perform "When the fellowship moves..." actions.
4) Move your player marker to the next site. (The fellowship's "current site (and region)" changes at this point).
5) Perform "When you move to..." actions.
6) Add twilight tokens equal to the new site's Shadow number.
7) Add 3 twilight tokens if the new site is in region 2; or 6 if it is in region 3.
8) Add 1 twilight token for each companion.
------------------------The fellowship is "at" the new site from this point on.----------------------
(So anything that checks the site the fellowship is currently "at" only applies after this point.)
<Fellowship or Regroup phase ends>