Is "connexion" synonymous with "connection"? For example, "an ethernet connexion". Can I use it like that?
Connexion is the original and variant spelling of "connection", common until at least the 19th century, and still occasionally used in British English (it was the house style of The Times of London as recently as the 1980s). It is derived from the Latin connexio, hence the spelling, unlike most words ending in "-ction" which are derived from Latin words ending in "-ctio" (e.g. "protection" from protectio).
I've never seen that spelling in use in the US.
In British English, connexion is an alternative spelling of connection; American English only uses connection.
The origin of the word connection explains the reason of this.
Connection has origin from the Latin connexio(n-); only in the 18th century the spelling -ct- started to be used, on the pattern of words like collect, and collection.
It is definitely used in British English. I came across this variant of spelling in Lord of the Rings: The Fellowship of the Ring. If we look closely at the author of the book, J.R.R Tolkien, he is a Briton and wrote this book in 1954. The usage of this spelling is dated and is closer related to 19th century/ early 20th century. In Canada, we spell it just like in the US: connection.
It mainly appears in (some) British texts and/or archaic texts. In use of typical American English, the word "connexion" would simply be replaced with the standard term, connection.
The term "connexion" is depreciated in modern American English. It is still (sometimes) used in British English.
An Example of Encountering the Term Connexion In a Modern Context
I found the use of the term "connexion" in some translations of Kant's Prolegomena to any Future Metaphysics.
Kant is still being read in philosophy today, so that is one use of the term "connexion" in a modern context.
This was the book:--
You can find the Kant quote below.
Sources and References:--
Other References and Further Reading:--
"I therefore first tried whether Hume's objection could not be put into a general form, and soon found that the concept of the connexion of cause and effect was by no means the only idea by which the understanding thinks the connexion of things a priori, but rather that metaphysics consists altogether of such connexions."
Emphasis via italics is by me and is not from the original text.
-- Kant, Immanuel. “Kant's Prolegomena: To Any Future Metaphysics (English Edition) Kindle Edition.” Kant's Prolegomena: To Any Future Metaphysics (English Edition) EBook: Immanuel Kant: Amazon.de: Kindle-Shop, https://www.amazon.com/Kants-Prolegomena-Future-Metaphysics-English-ebook/dp/B01LZ8UD4C.