What are the proper definitions of moment, couple, torque, 'moment of force'? When I read some scientific articles I am getting confused with the interchange being used between Torque, Couple, Moment, 'Moment Of Force'.
Have I got this correct?
Moment is supposed to be some action of a quantity at a distance from some point.
Couple are two equal and opposite forces acting on an object , but with a distance between them.
Torque is a twisting effect where a force is applied at a distance from some point on a physical object (I am assuming there must be some physical medium connecting that force to that point on a physical object).
Moment Of Force  = Torque
Moment of Couple =  Torque effect (cannot be a Torque because it's not a single force about a point)
Why do some say a Couple is a Torque?
Why do some say Torque is a free vector when they actually mean Couple?
Isn't there a need for the science community to provide better clarity on these definitions?
 A: As @Gandalf61 pointed out you can find a definition of torque on Wikipedia.
Although the terms moment and torque are often used interchangeably, since they are mathematically the same, a moment differs since it is used in connection with requirements for static equilibrium so that a moment does not actually cause rotation. It is only a measure of the tendency to cause rotation that must be counteracted by other moments so that rotation does not occur for equilibrium. In short, the term moment is used in statics whereas the term torque is used in dynamics.
Your understanding of a couple is basically correct but it is different from moment and torque because it involves two equal and opposite parallel forces that can cause rotation without translation. Note that the two forces must be parallel.
Hope this helps
A: Torque, moment and moment of force are often used interchangeably to mean the twisting effect of a single force about a given axis.
However, torque can also be used to mean the twisting effect of a pair of equal forces acting in opposite directions at different points i.e. a couple. With this meaning, the magnitude and direction of the couple’s torque is independent of where it is measured, hence it can be referred to as a “free” vector.
The Wikipedia article on torque explains these different usages clearly.
A: I like to define torque (or moment of force) as the work per unit angle of rotation that can be done by a force (or a combination of forces) acting in a manner that tends to cause a rotation.  This helps remind you that there is a distance involved (proportional to the radius) and that you need the component of force in the direction of motion. (And it's consistent with the relation: work = torque x angle.)
A: The term moment of X implies that X happens at a distance as you mentioned. Additionally, there is a commonality in how they are calculated which involves the cross product of position and X. The cross product is used to extract the moment arm distance to that X.

*

*Moment of rotation (aka velocity) => $\boldsymbol{v} = \boldsymbol{r} \times \boldsymbol{\omega}$

*Moment of momentum (aka angular momentum) => $\boldsymbol{L} = \boldsymbol{r} \times \boldsymbol{p}$

*Moment of force (aka just moment) => $\boldsymbol{M} = \boldsymbol{r} \times \boldsymbol{F}$
So if you want to be technically correct, use the moment of X terms, and not the colloquial ones such as velocity, angular momentum and moment. I know crazy!
But you cannot do that, because you can have velocity without rotation, or moment without a force. The velocity of a purely translating rigid body is not generated from rotation, but it is the same for all parts of the body. It is a free vector because it is not associated with particular location, like the moment of rotation.
Similarly, a pure torque is not generated from a force at a distance (and hence the term moment is avoided) but something felt equally by all parts of the body. It is also a free vector because it is not associated with a particular location, like the moment of force.
A common way to generate a pure torque is by a force couple (aka just couple) which means two equal and opposite forces offset from each other arranged in such a way to generate the specific torque vector needed. This is mostly a result of the fact the mechanics primarily deals with contacts between bodies which only forces at the contact points, and there is no good way to apply a pure torque to a body without applied some kind of force combination also.
In practice, torque is meant to be used when the result is known (a moment along a specified axis) but the means of generating this torque aren't important. But a moment is used when the details of how it is generated are important.
Consider the following example

A rotating shaft with an asymmetric mass attached to it is cantilevered off one end of the shaft with a bearing, and a torque is applied on the shaft. Find the reaction forces and moments on the bearing.

Here there is a distinction between the shaft torque whose details are unimportant to the problem other than the moment is along the shaft axis and the reaction moments of the bearing whose details are important and act along an unknown arbitrary direction.
