Speaking from the pov as a former astrophysicist, that definition of redshift is not quite accurate.
While we have observed the universe expanding since Edwin Hubble published his findings 90 years ago, it was in fact the Friedmann equations produced 10 years earlier that established how it would work within the context of general relativity.
The Universe is expanding at every known point at the same time at the same rate. However, this is only an observable effect over > 10 Mpc of distance (e.g. extra-galactic distances). This is far too large to effect a point particle like a photon no matter how far it travels so I'm afraid your analogy of a rubber sheet is incorrect.
The lengthening of said photon's wavelength comes instead from doppler expansion due to the nature of differing relativistic reference frames.
According to our local reference frame, doppler effects of nearby stars and galaxies are only due to local motion such as Andromeda being blue shifted as it approaches us for our eventual merger. However, as you go further and further out from our galaxy the metric expansion of spacetime begins to dominate the relative motion of all bodies, leading to observing progressively redder and redder light curves. A galaxy billions of light years away is perceived to be moving away from our reference frame at a significant percentage of the speed of light!
However, if you were to change our relativistic reference frame to match a distance galaxy's you would not see its photons "stretched out" but in fact they would look completely unshifted despite having traversed nearly all of spacetime, expanding as it went.