According to the Merriam Webster dictionary:
Moment - a very short period of time
Does the word "moment" in quantities like the moment of force or moment of inertia refer to this colloquial meaning of "moment" too? If so, where exactly is the "short period of time" involved in, say, moment of force? For me it has always been just the tendency of a force to rotate an object around an axis and I don't see any reference to time there.
The meaning you quote is only one of several. From the same dictionary, others which are now obsolete or not often used are
3: importance in influence or effect
4 obsolete : a cause or motive of action
and it is from these that the scientific meanings derive :
6a : tendency or measure of tendency to produce motion especially about a point or axis
6b : the product of quantity (as a force) and the distance to a particular axis or point
Definition 3 survives in the adjective "momentous" which is still in common use.
The moment of a vector $\vec v$ applied in the point $\vec p$ with respect to the pole $\vec q$ is by definition
$$\vec M = (\vec p - \vec q) \times \vec v$$
With $\times$ vector product.
In the language of physics, a moment is a physical quantity which accounts for how a physical property is located or arranged. We do NOT use the colloquial meaning of the term moment. This "moment" we are referring to is derived from the latin word momentum which is also a physical quantity equal to the product of the mass and velocity.
