Law's of Mechanics are Galilean invariant

The additional remark is not a corollary.

The expression 'galilean invariant' is abstract, which allows it to be short.

The text that follows is an attempt at stating what the expression 'galilean invariant' stands for.

That is surprisingly difficult; it is an attempt to express an abstract concept in concrete words. Whatever you try, it will always be awkward.

I prefer the following assertion of galilean invariance: (Yeah, it's longer.)
There is an equivalence class of inertial frames of reference. Each member of the equivalence class has a uniform velocity with respect to any other member of the equivalence class. A local mechanical experiment cannot find the magnitude of velocity with respect to any other member of the equivalence class.

Velocity is a vector quantity, so for any single member of the equivalence class you can think of the set of all velocites of the other members as a space. An abstract velocity space. But it is a space that doesn't have a zero point. When describing what galilean invariance stands for you have to accommodate that not-having-a-zero-point. You cannot single out a member; the very property that you cannot single out a member is fundamental.

You cannot tell whether you are moving with respect to something else

This is true with a proviso, that you are moving at constant velocity, here meaning that you aren't accelerating. This is the content of Galilean relativity. Which is why your later statement:

You cannot tell whether you are being accelerated wrt another frame or vice-versa

is false.

Hence:

That either can be moving wrt the other

is ambiguous, as you haven't specified how they are moving. That acceleration isn't Galilean invariant was partly demonstrated by Newtons bucket experiment where a bucket partially filled with water is rotated at constant speed. Here, we see that the surface of the water becomes concave and so thos motion can't be Galilean relative. Now, although it is rotating at constant speed, it's not constant linear velocity, and so this is acceleration and hence this is why Galilean relativity doesn't apply.

Galileian relativity tell us about inertial reference frames, systems in which Newton's second law is valid: $vec{F} = mvec{a} $. We are in the realm of Newtonian mechanincs.

This principle of relativity tell us that the law of physics are the same in all inertial reference frame (r.f), meaning that the structure of the equations are the same in systems that move with constant velocity one respect each other. Constant means that the magnitude and also the direction doesn't change.

This implies that if we have say three r.f: A, B and C; if A is moving at constant velocity with respect to r.f B and C, means that B are moving with constant velocity respect to C.

So, mechanical experiments performed in A, doesn't tell you if A is moving respect to B or C. You can immagine to do such an experiment like characterizing simple pendulum, or the collisions of balls, and you are closed in a boat or in a train moving at constant velocity respect to the ground. Without can't seeing outside, you cannot say if you are moving respect to the ground or not, watching the results of these experiments.