Hazelnut. You really are confusing things with your approach climb theories.
Lets just clear this up. There is no misnomer. Boeing clearly define two distinct and non interchangeable limits. They are called 1) Landing climb weight limit and 2) Approach climb limit weight. They are so clearly different that the limits are even separately shown and separately labelled on the landing details page of the OPT section (dispatch) of the electronic flight bag whenever you do a calculation. So again no misnomer.
Only the landing climb weight limit is displayed in the FCOM/QRH because only the landing climb weight is used when performing the overweight landing cx list. If you are ever landing over this limit or overweight on one engine the cx list tells you to configure the approach with a reduced flap setting (very similar to approach climb limit config ). Clearly inferring that on this aircraft landing climb weight limit is always a greater consideration than approach climb limit.
To suggest that somehow the entire FAR flight test schedule for demonstrating approach climb limits can be substituted for gear locked down performance is an interesting theory.