What does correcting for LSR mean in galactic velocities? This is a two-part question, but both parts are related:

I'm aware that modern astronomical software can correct velocities for the local standard of rest (LSR), but what does that really mean? As in, if you have measured a radial velocity by hand (e.g., from a plot of a spectral line) and you want to correct for LSR, how is this done?

As an exercise, I want to compare the galactic proper motions (which I presume are another way of saying velocities?) of a group of stars within 2kpc from the Sun between a method described in Eqs.30-32 of Mignard 2000 (Hipparcos data), and Eqs. 1-2 of Bovy 2017 (Gaia data). Apparently I also have to correct these velocities for LSR, but how is that done? Do you add a specific value? I thought these equations already include the components of the Solar motion w.r.t. to LSR, unless the constants included in the equations have nothing to do with that. I'm probably missing a big point here and books haven't helped me at all.

Thank you in advance.
Proper motions are not velocities. They are angle traversed per unit time. They can be converted to tangential velocities (i.e. tangential to a line joining the star and the Sun) if the distance to the star is known, but you need additional information to get the line of sight (radial) velocity and hence a 3D velocity.

Such velocities are with respect to the Sun (or possibly the solar system barycentre - a tiny difference).

The local standard of rest (LSR) is an estimate of the average motions of stars in the solar neighborhood. The Sun has a peculiar velocity with respect to the LSR. To get a velocity with respect to the LSR, you need to subtract the solar motion with respect to the LSR from your heliocentric velocities.

There are disagreements in the literature (at the level of 1 km/s) as to what the solar motion with respect to the LSR is.