In the two types of models, "exchange" is used in almost completely different ways. It looks like an unfortunate bit of linguistic history.
In the symmetry-based models, the "exchange" under consideration is entirely hypothetical. Suppose that your system has Larry the electron on the left, and Moe the electron on the right, but you turn your back for a little while and when you return the two particles have switched ("exchanged") places. Can you tell? The symmetries of your system --- that is, whether your system is the same after Larry and Moe switch places --- put constraints on the allowed wavefunctions. Most famously, a system made of identical fermions has to undergo a sign change if two identical particles swap places; this (anti-)symmetry gives rise to the Pauli exclusion principle and the solidity of matter.
This is interesting in the nuclear case because, in the limit where strong isospin is a good symmetry, the neutron and proton are different isospin projections of identical nucleons.
But nothing actually trades places in the symmetry-based "exchange" models; you're putting constraints on the allowed states so that it's impossible to tell whether such an exchange has taken place or not.
In the meson-exchange models, on the other hand, the "exchange" is less hypothetical. Here you have nucleons who are trading energy and momentum with each other via excitations in the meson field around them, which can be modeled as virtual particles obeying Bose-Einstein statistics. The meson-exchange models are very similar to quantum electrodynamics, where the interaction between electric charges is described by the "exchange" of virtual photons. Yukawa's insight was that if the force-mediating bosons are massive, then the potentials they produce will have a finite range.
Further muddying the waters, there is some overlap between the two types of interactions. For example, the pion can be thought of as an isospin-raising or -lowering operator acting on a nucleon. If Larry the neutron and Moe the proton exchange a neutral pion, they've retained their identities. If, instead, Larry the neutron and Moe the proton exchange a charged pion, the final state is as if they have swapped places, and the symmetry considerations come back into play.
As a practical matter, the purely symmetry-based models seem to come from earlier in the history of quantum mechanics, and the force-carrier models seem to be more modern improvements. The text to which you refer (published 1983)
seems to discuss the old symmetry-based models as a way to qualitatively justify the broad outlines of the nucleon-nucleon force, but includes a plot of nuclear potentials derived from the meson-exchange model.
Finally, the meson-exchange model has the flaw the the QCD meson spectrum is too complicated. There are an enormous number of possible force carriers, many of which are massive enough that the Yukawa-type interaction doesn't turn on until the nucleons are substantially overlapping --- in which case the validity of talking about separate nucleons exchanging a meson, rather than some many-quark interaction, gets called into question. The current trend is to discuss effective field theories where the nucleon-nucleon force happens at a pointlike vertex, and the number of such vertices to consider is restricted by the quantum numbers which do or don't change in the interaction.
