You're right in all of that.
But there is a further additional factor in the equation: Li-Ion batteries have different degrees of efficiency when delivering it's stored energy depending on the discharge rate, and that affects it's perceived capacity (in mAh).
source: Battery Performance Characteristics - How to specify and test a battery
Since different suppliers can test their bats at different discharge rates (typically at very low discharge rates), the mAh ratings between different bats may not mean exactly the same thing.
But there is a further additional factor in the equation: Li-Ion batteries have different degrees of efficiency when delivering it's stored energy depending on the discharge rate, and that affects it's perceived capacity (in mAh).

source: Battery Performance Characteristics - How to specify and test a battery
Since different suppliers can test their bats at different discharge rates (typically at very low discharge rates), the mAh ratings between different bats may not mean exactly the same thing.