How could inertial and gravitational mass be even conceptually different?

Question

From https://en.wikipedia.org/wiki/Mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. [...]
Suppose an object has inertial and gravitational masses $m$ and $M$, respectively. If the only force acting on the object comes from a gravitational field $g$, the force on the object is:
$$F=Mg.$$
Given this force, the acceleration of the object can be determined by Newton's second law:
$$F=ma.$$

In theory, mass could be determined by the number of indivisible particles the object is made of. A better approach would be choosing unit mass and using law of conservation of momentum:
$$frac{m_1}{m_2}=-frac{Delta v_2}{Delta v_1}.$$
If mass can be determined in the absence of any force, how could there (even conceptually) exist more types of mass?
Shouldn't we talk about "inertial and gravitational force equivalence" instead of about "inertial and gravitational mass equivalence"? Any kind of mass which is "not invariant" under different kinds of forces makes no sense to me.
The answers here (Why did we expect gravitational mass and inertial mass to be different?) and here (Question about inertial mass and gravitational mass) do not answer my question as mass is "determined" by force there.

Dale · Accepted Answer

In theory, mass could be determined by the number of indivisible particles the object is made of.

This would only be the case if all indivisible particles were the same mass and if the mass of a composite object were equal to the sum of the masses of the indivisible particles. Neither of those are true. Mass cannot be determined this way.

A better approach would be choosing unit mass and using law of conservation of momentum:
?1/?2=−Δ?2/Δ?1

Note that for this to work requires $Delta v_1 ne 0$. That in turn requires a force. So this method does not avoid the need for a force. However, what it does do is make it clear that the resulting measure is independent of the type of force, without eliminating the need for a force altogether. Of course, the force based definitions also do that, but not as clearly.

Shouldn't we talk about "inertial and gravitational force equivalence" instead of about "inertial and gravitational mass equivalence"?

Probably if we ever found gravitational mass to be different from inertial mass we would call it gravitational charge instead. So mass would continue to refer to the inertial mass. Then, just like the acceleration of an object in an electric field depends on the ratio of its electric charge and mass, so also the acceleration of an object in a gravitational field would depend on the ratio of its gravitational charge and mass.

Dims · Answer

Gravitational mass could be completely different quantity, why not? We have electrical charge, which behaves very similiar to gravitational mass, but is something different. We can have two bodies with same mass but dirrefent charge.
In other reality it could be more variants.
Suppose bodies A and B. You put them on rails and tried to push. You found that it is equaly hard to push them. This meant that these bodies have identical inertial masses. Now you put these bodies onto scales and saw, body A is heavier than body B. That meant that Earth attracts body A stronger.
This is rather possible picture in alternative reality, why not?

How could inertial and gravitational mass be even conceptually different?

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