Is “fill in” a British verb?

In the ever popular British versus American English language Binary, the phrasal verbs “fill in” and “fill out” are 0ften mapped nationally. A cursory Google search will produce a legion of results that claim “fill in” is a British phrase and “fill out” is American. But as with any binary, this is merely a glancing blow to the complexity of the underlying usage and distribution.

section from The American Educator Volume 3 published in 1897. Text gives the definitions of the verbs "fill in": to insert or enter, and "fill out" : to enlarge.
The American Educator
A Library of Universal Knowledge: Volume 3 (1897)

For many speakers of American English the verb “fill out” means to complete something, following a common meaning of the preposition “out” in phrasal verbs such as “burn out”, “run out” and “work out”. Thus, these speakers would use the verb “fill out” for completing documents, but if they started completing the document and did not finish they would use the verb “to fill in”.

As a speaker of Standard American English, white Northerner subtype, I would say “I filled in the top of the form, you can do the rest” but “I filled out the form, what do I do now?”. I would also say “fill in the blank” and “fill in a hole” and “fill in the details”. The sentence “fill out the form by filling in the blanks” works perfectly in my variety of English.

Phrasal verbs are semantic units with multiple parts: a verb, a particle and sometimes a preposition. The meaning of each of these parts varies, giving rise to deep variation in phrasal verbs across and within Englishes . “Fill out” to many American English speakers entails an element of completion that it does not for speakers of other varieties of English. Whereas for speakers of British English the preposition “off” sometimes conveys a similar element of finality, as in the sentence “the event was rained off” , which would usually be rendered as “rained out” in American English.

The difference between “get on with” and “get along with” is another oft-observed difference, and it similarly owes its distribution to the meaning of the particle. In many varieties of UK and Irish English “along” is a particle that only indicates direction, as in the phrasal verbs “move along”, “pass along” and “run along”. For other speakers, however, the particle “along” has evolved a broader sense of “together” which is often used with the preposition “with”, as in the phrasal verbs “sing along”, “take along” and “play along”.

Section from William Howell's An Institution of General History published in 1680 that shows the phrase "he took his leave, taking what he could get along with him".
Howell, William: An Institution of General History, Or, The History of the World, (1680).

Phrasal verbs are complex interlocking structures and they vary across many different features. While it is easy to imagine and enjoy a simple national distribution of language variation, the bigger picture is a lot more fascinating.