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In my opinion there is every reason to resist the flux-based definition.
As it turns out the flux-based formula is the result of applying the
real definition to a linear inductor with geometry and electrical
characteristics which do not vary in time.
The "swinging choke" comes to mind immediately, of course,
but astrophysical examples abound as well.
While I'm railing against these neosacred cows, let me also deplore the
use of the word "potential (meaning electrical potential) in this or
any other time dependent application. I like to firmly associate the
terms "potential" and "time invariant" with one another. Perhaps "emf"
would be a better choice?