I note there are two terms about Galactic rest frame in literatures: Galactic Standard of Rest (GSR) and Galactic Rest Frame (Grf).
Are they the same frame? If not, what's the difference?
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They're the same.
Local Standard of Rest (LSR)
Let me start out by defining the dynamical Local Standard of Rest (dynamical LSR or also LSRD). This definition is taken straight from An Introduction to Modern Astrophysics, 2nd Ed. by Carrol & Ostlie, page 903 (emphasis theirs).
To investigate the motion of the Sun and other stars in the solar neighborhood, we first define the dynamical local standard of rest (dynamical LSR) to be a point that is instantaneously centered on the Sun and moving in a perfectly circular orbit along the solar circle about the Galactic center.
Note that once the LSR is chosen, the Sun immediately begins to drift away from it, implying that we would effectively need to redefine the reference point constantly. In reality this is not a significant problem because (fortunately) the 230-Myr orbital period of the LSR is very long compared to the time since modern telescopic observations began (and even longer compared to the lifetime of a typical research grant); consequently there has not been sufficient time for the effect to become noticeable.
Galactic Standard of Rest (GSR)
Now it is important to understand how the LSR coordinate frame is defined, because the Galactic Standard of Rest (GSR) is defined with respect to the LSR. You can read about it here, but in effect, the LSR is a non-inertial coordinate frame co-rotating with the rotation of the galaxy. The GSR is defined in the same way, but removes that co-rotation by subtracting out the rotational velocity of $220\:km/s$. In the LSR frame, another star's velocity would essentially be with respect to our Sun but in the GSR, the star's velocity would be with respect to a fixed point in the galaxy and be a summation of that star's motion with respect to our Sun and it's orbital motion around the galaxy.
Galactic Rest Frame (GRF)
The GRF is identical to the GSR, although I've heard GSR more often. In this paper, you can see they define the GRF to be
... the galactic-rest-frame (grf) proper motions ... which are those seen by an observer at the location of the Sun and at rest with respect to the Galactic center.
This is identical to the definition of the GSR above. Another paper similarly defines the GRF.
Removing the contribution of the motion of the Sun and of the Local Standard of Rest to the measured proper motion produces a Galactic rest-frame proper motion.