Canonical vs. Noether momentum for longitudinal waves on a 1D chain

Consider $\phi:x\rightarrow R$, to be a time dependent map from the position space labeled by $x$ or $i$ to a 1 dimensional real line that I'll call the 'target space'. Since you are talking about periodic boundary conditions $\phi$ traces out a loop in the target space as you vary $x$ (it is a loop embedded in 1 dimension so it retraces its path).

This loop is localized in the target space. You can define a `center of mass' at a mean position $\phi_0$. There are then two translation symmetries. You can shift the $\phi\rightarrow \phi+a$ which shifts the position in the target space. The conjugate momentum $\Pi$ is the momentum of the center of mass in the target space, and as you saw you can define a boost operation on $\Pi$.

The other symmetry is translating in the $x$ or $i$ space. As people have pointed out in the comments this symmetry also exists in the discrete model and has nothing to do with the continuum limit. The conjugate momentum $P$ describes the flow of energy around the loop for fixed center of mass in the target space.

There is another hidden conserved quantity if $\phi$ is considered to map to a compact target space like a circle instead of the real line (i.e. $\phi$ is an angular coordinate). Then the loop can fully wind around the target space before it comes back to its starting point and the number of windings $m$ is conserved with time along with $\Pi$ and $P$.

As you might have guessed this has a connection with string theory. Fields like $\phi$ describe coordinates in the target space and the two coordinates $x$ and $t$ that label points along the loop of string are called coordinates of the 'world sheet'.

Back to longitudinal springs

It occured to me you might want some more down to earth explanation in terms of longitudinal springs. There is a peculiarity here in the sense that the target space position $\phi$ appears to be existing in the same space as the internal word sheet position $x$. But note that you are using periodic boundary conditions in order to say it is translationally invariant in x. There is in effect a spring connecting the right and left endpoints in our $x$ label so these two endpoints want to be close to each other and the whole system forms a loop just as I described above. So $P$ still describes an internal energy flow along the chain of springs.

And note that your derivation of the Lagrangian is still valid in a boosted frame where all springs are moving with the same forward velocity, so $\Pi$ still describes the momentum of the center of mass of the entire system as opposed to the internal energy flow around the loop.

If you compactify your overall $\phi$ position with periodic boundary conditions then you can also talk about the conserved number of windings $m$ as I mentioned above. But note these periodic boundary conditions are distinct from the periodic boundary conditions in $x$ which amounts to adding an extra spring between the endpoints.