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What is the definition of inertial mass? I can see two options, either it's the coefficient associated with the object being accelerated in Newton's 2nd Law, or it's the coefficient relating momentum to velocity. Or they are equivalent.

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Well, if it is constant, then it's equivalent.

Inertial mass is the resistance that a body offers when a force tries to change its state of movement.

So it is the $m$ that appears here

$$p=m\dot{x}$$

and here

$$F=m\ddot{x}.$$

But if it's not constant you can see it as the magnitude that appears in

$$ p=m\dot{x} $$

because in the force expression, there is a new term due to the mass variation. But the definition is the same.

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