Below is a portion of the page contents from https://www.grammarphobia.com/blog/2016/01/extrovert-extrave... to discuss this issue.
Extrovert or extravert? January 11, 2016
Q: I make a point of using “extravert,” not “extrovert,” because that’s how Myers-Briggs spells it. I did the personality test and learned I’m neither an “introvert” nor an “extravert.” I’m right on the line—I call myself an “ambivert.” Your thoughts?
A: We’ve checked six standard dictionaries and all of them list “extrovert” as the principal spelling for someone with an outgoing or gregarious personality, though five include “extravert” as an acceptable variant. The two spellings showed up in writing at about the same time, “extravert” in 1916 and “extrovert” in 1918, according to citations in the Oxford English Dictionary. Etymologically, “extravert” is the term one would expect. In Latin, extra means outside and vertere means to turn. So an “extravert” turns outward.
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Despite the questionable etymology of “extrovert,” speakers of English overwhelming prefer it to “extravert,” which explains why “extrovert” is the principal spelling in standard dictionaries. The term “extravert” is more at home in the literature of psychology. That’s why the Myers-Briggs Type Indicator, the personality questionnaire you filled out, lists “extraversion,” not “extroversion,” as a psychological preference. As Oxford Dictionaries online explains, “The original spelling extravert is now rare in general use but is found in technical use in psychology.”
In fact, standard dictionaries generally define the term one way in the language of psychology and another way in common usage. In psychology, according to the online Merriam-Webster’s Unabridged, it refers to “one whose attention and interests are directed wholly or predominantly toward what is outside the self.” In general usage, however, it simply refers to “a gregarious and unreserved person,” Merriam-Webster’s says. However, the principal spelling in dictionaries is “extrovert,” whether the word is used in the psychological or the general sense.
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