Edit: I think your discovery fits into an existing terminological framework (though this does not give much of an explanation):

• the iteration you defined is an $n$-color totalistic cellular automaton with, however, non-periodic boundary conditions.

Remarks.

• Note that your rule that the vertex-to-be-updated does not contribute its current value to the sum which determines its next state can be 'simulated'/'is an instance of' the definition of 'totalistic cellular automaton' on this website. I am only speaking about your question for paths; and I take it that you are working over the ring $\mathbb{Z}/(n)$, where $(n)$ denotes the principal ideal generated by $n$.

I assume you are working with the path graph $0\text{-}1\text{-}\dotsm\text{-}n-1$.

I write $v_i^{t}$ for the value of vertex $i$ at time $t$.

For $0<i<n-1$ you require¹

$\mathbf{v_{i}^{t+1} + (n)} = v_{i-1}^{t}+v_{i+1}^{t} + (n)$ ${}\qquad\qquad$ (global.rule)

At the boundaries you require (which is a serious complication when it comes to making a precise connection with the existing literature on cellular automata)

$\mathbf{v_{0}^{t+1} + (n)} = v_{1}^{t} + (n)$ ${}\qquad\qquad$(left.boundary.rule)

$\mathbf{v_{n-1}^{t+1} + (n)} = v_{n-2}^{t} + (n)$ ${}\qquad\qquad$(right.boundary.rule)

Now the point I would like to make is that your (global.rule) just is one of the hundreds upon hundreds of rules known and documented by Wolfram. I did not try to find out which decimal number they have assigned to your rule. Nor did I try to find out whether Wolfram has a classification for rules which suddenly change at the boundaries (such as yours do). Let me just mention that this violates what

Carsten Marr, Marc-Thorsten Hütte: Outer-totalistic cellular automata on graphs. Physics Letters A 373 (2009) 546-549

in Section 2 call 'Homogeneity'.

Let me also mention that looking into K. Salman: ANALYSIS OF ELEMENTARY CELLULAR AUTOMATA BOUNDARY CONDITIONS,

and contacting the author, seems a relevant thing to try.

¹ _{I use the boldface for readability only; it is not inteded to have mathematical meaning.}