A period of fighting is a round or more of fighting - the kind supported by the rules.
My understanding of your interpretation of the text is that it doesn’t say anything about a period of fighting. It just says that “fighting” is a period of strenuous activity.
A period of spell casting is the casting time of the spell.
Same as above with the added complication that casting time isn’t given in periods of time at all, but in units of the action economy, which aren’t periods of time.
The rhetorical gambit of claiming those are not periods is the weakest of your arguments for your interpretation.
My claim isn’t rhetorical, and it isn’t meant to support my interpretation. It’s meant to point out the flaw in yours. I’ll explain it as clearly as I can.
Here’s the relevant quote: “a period of strenuous activity — at least 1 hour of walking, fighting, casting spells, or similar adventuring activity”.
If each item separated by commas after the em dash independently corresponds to “a period of strenuous activity”, then we should be able to remove one or more of the items without significantly altering what the list is defining. Let’s see how it goes:
a period of strenuous activity — at least 1 hour of walking
So far so good. The item “at least 1 hour of walking” clearly corresponds to “a period (at least 1 hour) of strenuous activity (walking)”. Now let’s try it with the items for which no qualifying length of time is given:
a period of strenuous activity — fighting, casting spells, or similar adventuring activity
In this version, the list of items only corresponds to types of strenuous activity, changing the function of the list in the sentence from defining “a period of strenuous activity” to defining “strenuous activity”. I find this inconsistency to be a major flaw with your interpretation.