Is the definition of Work done by Torque a general result?

I was going through this post.

In the post the person who asked the question mentioned a statement:

"Lastly, I am emphasizing that I’m am NOT saying that torque is incorrect, only that the work derivation of torque (which I believe is very common) is invalid and that there must be another explanation.Lastly, I am emphasizing that I’m am NOT saying that torque is incorrect, only that the work derivation of torque (which I believe is very common) is invalid and that there must be another explanation."

This got me to wondering. It is very common to find the definition of Work done by Torque in Undergrad books as
$$dW = vec tau . dvectheta tag1$$

However I think that the above definition is only valid or at least useful when the acceleration of centre of mass of a system is zero.

And in case where system’s centre of mass has an acceleration, it is better to use the general definition of Work as

$$dW = vec F . dvec r tag2$$

where $dvec r$ represents the displacement of point of application of force $vec F$.

However I am unable to understand that if equations $(1)$ and $(2)$ are synonymous.

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Question:

1. Is equation $(1)$ strictly a general result or only applies to certain special cases?
2. Are equations $(1)$ and $(2)$ same?