The meaning of keeping the bare parameters fixed

So, this question concerns two different kinds of renormalization group equations. I would like some clarifications, if possible.

1. The usual RG equations taught in QFT courses, like the Callan-Symanzik or t'Hooft-Weinberg equations: Here the bare couplings of the theory are kept fixed.

2. The RG equations used when studying the continuum limit of lattice theories: Here the renormalized parameters of the theory are kept fixed.

For concreteness let us think about a theory with a relevant mass parameter $m$ and a marginal coupling $g$.

Now, the physical meaning of the second type of equation seems intuitively clear to me, it describes how I have to tune the bare $m_0$ and $g_0$ as I take a higher and higher cut-off, to keep the IR physics (i.e. a mass and a low energy scattering cross section) fixed. Moreover the equations tell me how the Green functions behave as I approach the continuum limit.

(Comment: I know that in general the renormalized mass is not the same as the physical mass (defined with the pole of the propagator), but If you define the renormalized mass appropriately, than close to the continuum limit they will be close, and will have the same continuum limit, so this prescription actually keeps the IR physics fixed.)

However, it is much harder for me to make sense of the first (more common kind) of RG equation. If the theory has a natural cut-off, like the inverse lattice spacing in a theory of condensed matter physics, than I can imagine, that defining the theory with the bare parameters, i.e. the actual microscopical physical parameters, makes sense, and keeping the bare parameters fixed actually guarantees that everything the theory predicts remains fixed. The physics remains fixed. However, this kind of interpretation does not seem natural to me in particle physics. There, the cut-off is not physical, and to get physically meaningful predictions, in the form of a renormalized perturbation theory, one needs to tune the unphysical bare couplings in a way that they cancel (to that order) the unphysical cut-off choice I have. So why do we derive the RG equations taking the bare couplings to be fixed?