Eric the Orange said:
There's a third term, which you are missing and which is needed to complete the set.
Gender Identity
Gender identity is the personal sense of one's own gender. Gender identity can correlate with assigned sex at birth or can differ from it. All societies have a set of gender categories that can serve as the basis of the formation of a person's social identity in relation to other members of society.
When you say you are male, you aren't referring to the gender norms of the society you live in. Instead, you are either referring to a personal sense of "belonging" to a given gender (a personal identity) or to the way you see yourself as socially positioned in relation to the society you live in (a social identity). In all likelihood, you're referring to both at once.
People still sometimes talk about gender norms as if they were universal constants of any given society, as if any society will have a single male and female "role", but that isn't actually true. Even if you're deeply conservative, your idea of how it is normal for a man to behave will be different from other men. To a certain extent, breaking gender norms isn't a huge deal. Everyone will break someone else's gender norms (and usually their own gender norms) at some point in their life. Heck, when you actually start to break down the messages about what the "gender norms" of our society are, you'll find they're riddled with contradictions and ridiculous idealizations which make them impossible to actually adhere to in entirety.
The reason we can square this is that, as humans beings, we all understand a difference between what we do and who we are. We know that behaving in "unmanly" ways doesn't actually change our gender identity, either in personal identity or how society percieves us. We also understand, on some level, that our gender identity is more than our bodies. When people lose their sexual characteristics due to disease or injury, their gender identity doesn't magically change, instead they are more likely to develop dysphoria, or a distressing awareness that their bodies don't align with how they percieve their "true" selves.
In conclusion, the reason you are seeing similarities between these terms is because they are actually part of a taxonomy. Gender is a broad category referring to all the societal information, culture and baggage around sex. Within the category of gender, there are gender norms, which are the sociological forces that tell us how gendered people
should act and behave, but there are other related concepts too, like gender identity, which is the psychological consciousness of being or belonging to a particularly gender.
In fact, bridging the two is another concept, gender expression, which is the features by which a person socially signals their gender identity to the world by referencing gender norms. Personal identity becomes a social identity which is expressed using the "language" of norms, and all of these things fall under the umbrella concept of gender.