The Stacks project

I suppose what you really want in this definition is whatever is needed for MacLane's coherence. It might be worth mentioning that the other minimal coherence conditions: $(1\otimes X)\otimes Y\to X\otimes Y$ equals $(1\otimes X)\otimes Y\to 1\otimes (X\otimes Y)\to X\otimes Y$, $(X\otimes Y)\otimes 1\to X\otimes Y$ equals $(X\otimes Y)\otimes 1\to X\otimes (Y\otimes 1)\to X\otimes Y$, left unitor $1\otimes 1\to 1$ equals right unitor $1\otimes 1\to 1$, $(1\otimes X)\otimes 1\to X\otimes 1\to X$ equals $(1\otimes X)\otimes 1\to 1\otimes X\to X$, and so on are consequence of the definition. I think the first three are not immediate, this proof is written in ncatlab, apparently it's due to Kelly https://ncatlab.org/nlab/show/monoidal+category. The others follow from naturality of unitors and associator, and one of them is the pentagon identity. In the case of symmetric monoidal categories you also want $1\otimes X\to X$ equals $1\otimes X\to X\otimes 1\to X$ . I believe a similar proof as the Kelly one works here using the hexagon:

$$\matrix{ X \otimes (1\otimes 1)) & \to & (1\otimes 1)\otimes (X) \cr \downarrow && \downarrow \cr 1\otimes X & \to & X \otimes 1 }$$

conmutes by naturality of conmutator

Up to associativity (as in MacLane Coherence) $X\otimes 1\otimes 1\to 1\otimes 1\otimes X$ equals $X\otimes 1\otimes 1\to 1\otimes X\otimes 1\to 1\otimes 1\otimes X$ (this is the Hexagon).

By naturality of the unitor $$\matrix{ 1 \otimes X\otimes 1 & \to & 1\otimes 1\otimes X \cr \downarrow && \downarrow \cr X\otimes 1 & \to & 1 \otimes X }$$

conmutes

As the arrows are isomorphisms this implies $X\otimes 1\otimes 1\to 1\otimes X\otimes 1\to X\otimes 1$ coincides with $X\otimes 1\otimes 1\to X\otimes 1$. As $(.) \otimes 1$ is an equivalence conclude that $X\otimes 1\to 1\otimes X\to X$ coincides with $X\otimes 1\to 1$.

These three conditions for monoidal categories and this extra condition for symmetric monoidal category I guess are straghtforward verifications in applications, so it's also possible to just add it to the definition instead.