I’m not sure I understand the description exactly.
For current, the measurement is inline. The circuit will basically be something like this:
A B C
+---------------------------+-------[resistor]-----------+
|+ | |
battery |F LED
|- | |
+--------------------------Board-------------------------+
D E
(Note that LED and resistor may be swapped, the result is the same.)
The multimeter can be “inserted” at A, B, C, D, E or F
A and D measures board + LED
B, C and E measures only the LED
F would only measure the board.
For voltage you don’t measure inline, but between two points, and those points should be C and E. If you insert a current meter at one of those points, make sure to measure the voltage of the diode on the side of the current meter that is closest to the LED to get the best accuracy. In general, the closer to the LED you get the better.
Please note; I think you’re already doing all of this right, I’m just trying to get complete clarity here.