I'm an extrovert (happy speaking publicly, I think out loud and so on) but not by nature particularly sociable. At least one good friend is the opposite.
Introversion / extroversion is a way of interacting and thinking, not a measure of desire to interact with people socially.
Extroversion is the characteristic of being outgoing and comfortable in social interactions. It has nothing to do with speaking publicly which is more about controlling performance anxiety, many actors and other performers are introverted in private life.
People who self declare to be "not by nature particularly sociable" may be introverts, some of them may be social recluses, what they aren't is extrovert.
But in terms of people who self declare themselves "not by nature particularly sociable" not being extroverted, I disagree and offer myself up as an example.
I am an extrovert. Psychological testing confirms this (and I'm a strong extrovert, not just marginal and this has been tested more than once) and no-one who has ever met me or worked with me would suggest anything else.
But I am also not by nature particularly sociable. I am an only child and by default prefer my own company (or that of a small number of close, established friends). I could happily spend days on end alone and often turn down the chance to go out because I don't particularly like the experience.
Extroversion is a way of thinking. While it may correlate with sociability, being an extrovert and being anti-sociable are in no way mutually exclusive.