I've been told that the use of ellipsis in "$S = x_1 + x_2 + x_3 + x_4 + \dots$" is ambiguous and meaningless. Is it? What else can the definition of the ellipsis symbol, "$\dots$", mean in this context?

$$S = x_1 + x_2 + x_3 + x_4 + \dots$$

All I can see is that you have an infinite sum of $x$s, where the first one is $x_1$, the next is $x_2$, then $x_3$, and so on forever, for as many natural numbers as exist and in order. But, for some reason, I am being told that such a definition is ambiguous and meaningless compared to formal mathematics.
Stating ambiguity of this summation is bad faith. The pattern is obvious and the ellipsis clearly indicates an unlimited sequence of terms.



I would be more critical towards a sum like

$$1+2+4+\cdots\ ?$$



In common practice, if the first few terms/indexes (as little as $3$) follow an arithmetic progression, it can be considered implied.
To give you an idea of what can go wrong with arbitrary entries in the summation the following example shows these complications. Begin with $S=1 + 2 + 4 + 8 + ...$ and so $S=1+2(1+2+4+8+...)$ and so $S=1+2S$ giving us finally that $S=-1$ and we now have a sum of positive numbers equal to a negative number, which is absurd. These complications must be addressed in order to deal with sums of this kind meaningfully.