"But by that argument, would not "gender roles" fall under the same definition of gender as opposed to sex as used above?"
No. Because Sex Roles and Gender Roles are terms that have been, and still are, used interchangeably, and mean the same thing.
Although because of the rampant snowflakism the term sex roles is probably now hater mchating hate speech. Not to mention the concept of gender roles in general...
" If someone whose biological sex is male, yet they identify and live as a female, regardless of what anyone else may think of the subject of being transgendered, then their "gender role" would fall under the female category of your argument."
The problem with your argument is that you're looking at how things are changing NOW. I'm saying that historically gender and sex had been used interchangeably, and both were understood to differentiate between those of the elephant and castle class vs the utero class.
Obviously in today's current usage both terms break down because of the... well, I was going to say fluid definitions of sex/gender, but I'm just going to say because of the rampant fuckstickery in which we now have about 7 billion "defined" genders/sexes, but if you actually SAY that there are about 7 billion "defined" genders/sexes you're a horrible, horrible cis normative hater/ahole because you're attempting to slot people in concretely defined terms.
It's a mad house.