If I understand you correctly, I think there are actually two points being asked about here, and so it will be beneficial to try and separate them.
First there is a question about the relation of the parameter $m$ which appears in the Lagrangian to the masses of physical particles. And there is also a question about how the identification of a "virtual"-like structure in a scattering process should be identified with a particle of specific mass.
Let me first tackle the second question. Forgetting everything about Feynman diagrams and perturbation theory, at the end of the day we measure scattering amplitudes which are merely approximated by a sum of Feynman diagrams. It's a theorem (proven in various QFT books. I think there may also be some more down to earth discussion in Griffiths particles book, and of course the most complete proof likely appears in Weinberg QFT volume 1) that resonances in the scattering momenta (which are poles in complex momenta) always correspond to particles in our Hilbert space. Furthermore, these particles need not correspond to the "fundamental" fields we write down in our Lagrangian. For example, the pion is a composite particle, but there still exists a pion state in the Hilbert space with definite 4-momenta, and hence certain scattering processes will have resonances at the pion mass despite there being no pion field in the standard model. This relation between poles and masses is also the reason why masses of particles are sometimes referred to as the "pole-mass" of the particle, just to distinguish from the mass parameters that may appear in the Lagrangian. (As a bonus, while the real part of the pole location is related to the mass, the complex part is related to the particle lifetime).
Now, if the Feynman diagrams are a good approximation to these scattering amplitudes, it may be the case that the tree-level diagram alone is a good approximation. If the tree-level diagram has a pole, then to within the precision of that approximation, so will the scattering process and hence there will exist a particle in the theory with approximately the pole mass, up to subtleties of renormalization.
Speaking of, I will point out that my statements about pole masses are completely non-perturbative, while may statements about tree-level approximations are perturbative. When we deal with diagramatic expansions, as soon as we want to go past tree-level and involve a look diagram we must renormalize all our couplings, including the mass parameters that appear in the Lagrangian. This will necessarily shift the locations of said poles.
A complete picture of how the non-perturbative picture, diagram expansions, and renormalization all tie together can be understood via the effective action. I think it would be too much to launch into a discussion of these things here, but I will note that there is a fairly nice description in the QFT book by V. Parameswaran Nair, though it may be some effort to parse. A much quicker route (but with correspondingly less detail) would be the QFT book by Thomas Banks.