There is no conflicting definition. Minutes, seconds, thirds, and fourths etc. are sexagesimal subdivisions, as tenths, hundreds, thousandths, ten thiusandths etc. are for decimal.
I know what you mean, and you are correct in that this is why we have minutes and seconds of arc, but the linked page is literally suggesting a different definition.
there is no multiple definitions going on: what happened is an _elision_.
The actual word is "second minute" (as opposed to the "first minute").
Most languages have by now elided "first" from "first minute" resulting in the "minute" as we know it today, and elided "minute" from "second minute" resulting in the "second" as we know it today.
i.e. "second" literally means "the one that comes after the first", but is implied to be about the subdivision of the small unit of time.
Part of the point is the "second" definition based on a subdivision is no longer the SI second. You are right about the elision, but in the context of the time most people are now eliding System International ("SI") from the beginning rather than "minute" at the end.
"Seconds were once derived by dividing astronomical events into smaller parts, with the International System of Units (SI) at one time defining the second as a fraction of the mean solar day and later relating it to the tropical year. This changed in 1967, when the second was redefined as the duration of 9,192,631,770 energy transitions of the cesium atom."
A while "prime minute" generally contains "61" "second minutes" it sometimes can contain 61 "second minutes"; so clearly the "second minute" is the true unit here :-)
