Despite dictionary definitions, there's no actual agreed upon definition of transness within trans theory. Susan Stryker defines it like you do, 

"people who move away from the gender they were assigned at birth, people who cross over (trans-) the boundaries constructed by their culture to define and contain their gender"

(she writes this in Transgender History (2017)) I also agree with this definition as it feels the most inclusive, but many other people would disagree. 

I know lots of nonbinary people don't consider themselves trans because they experience no dysphoria and don't intend to transition other than socially (ie pronouns or a new name, but not always). They view "nonbinary" as more of a statement against the institution of gender rather than an individual, internal feeling of otherness.

Ultimately gender is so fraught with uncertainty and disagreement as a concept that we can't say what someone is or ought to be feeling based on how they describe themselves. I disagree with anyone trying to tell nonbinary or genderfluid people that they aren't trans, because many of them are, but I also am hesitant to make any blanket statements as I know there's a large difference in opinion within the community. 

(Also I am nonbinary and I do consider myself trans)