I just got my first look at a borrowed copy of Nourse's new
"solve everything" book, specifically to look at the barrel puzzle
solution he gives. His notation is:

The two rotating pieces are "rings".
They rotate through 5 "notches".

"Layers" (horizontal slices containing 5 balls each) are
numbered 1 thru 4 from bottom to top, plus T for the 3
balls immediately under "plungers" (when they are up).
No notation is used for positions within the puzzle,
except this numbering of the layers. Positions are given
by word descriptions or drawings.

Moves:

Up-arrow move the plungers up
Down-arrow move the plungers down
Two boxes, one above the other, which represent movement
of the two rings. A "T" above, or inverted "T" below the boxes
shows the current plunger position. Each box may contain:
blank don't move this layer at all
Left-arrow rotate the ring left (clockwise, as seen from the top) one
notch
Left-arrow 2 rotate the ring left two notches
Right-arrow rotate the ring right one notch
2 Right-arrow rotate the ring right two notches

All moves assume the lonesome plunger is toward the puzzle solver.
All move sequences begin assuming the plungers are in the up position.

There is a great deal of room for improving this notation.
Particularly, the boxes should be dispensed with in favor of
some single-printed-line way of differentiating between top
and bottom rings. The current plunger position is not needed.
Because many terminals or typewriters do not have four
directional arrows in their character sets, these should be
replaced. I like thinking in terms of left and right rotations
rather than clockwise, since this puzzle is usually solved
without turning the puzzle as a whole. Position notations
are needed. For these reasons I propose the following:

Move notations:

U move the plungers Up
D move the plungers Down
T rotate the Top ring right (counter-clockwise from the top) one notch
B rotate Bottom ring right one notch
-T rotate Top ring left one notch
-B rotate Bottom ring left one notch
T2, B2, -T2, -B2 two notch versions of the above four respectively

Positional notation:

Same naming for "rings", "layers", "notches", "plungers",
except call the top layer "5".

The ball positions in layer one are, clockwise, as seen from
the top, beginning with the one under the lonesome plunger:
1F (front), 1L (left), 1LL, 1RR, 1R.

Correspondlingly for the other layers.
There is no 5L or 5R, because of the orientaion of the puzzle.

During a move, at times when the plungers are down, there
will temporarily exist positions 0F, 0RR, 0LL.

Any suggestions or arguments about the notation are welcome.

Nourse's solution has two typos in it. Page 33: upper rightmost
move on page should be in the bottom ring, not top. Page 36:
second move on the page should have a 2, not single notch.

If people are interested, I can summarize Nourse's approach to solving.

/Don Lynn