You are sure hung up on that word, "hunting"...

I think your definition and TPWD's definition may differ a bit.

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Hunt: To capture, trap, take, or kill, and includes the act of attempting to capture, trap, take, or kill.


https://tpwd.texas.gov/regulations/outdoor-annual/hunting/general-regulations/definitions

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A hunting license is required of any person (resident or nonresident), of any age, to hunt any animal, bird, frog or turtle in this state, except the following:

coyotes, if the coyotes are attacking, about to attack, or have recently attacked livestock, domestic animals, or fowl.
depredating feral hogs, if a landowner (resident or non- resident) or landowner’s agent or lessee is taking feral hogs causing depredation on the landowner’s land.
fur-bearing animals, if the hunter possesses a commercial trapper’s license.
depredating fur-bearing animals, if a landowner (resident or non-resident) or landowner’s agent is taking fur-bearing animals causing depredation on the landowner’s land. No portion of the nuisance fur-bearer may be possessed, sold or retained for any purpose.


"But Mr. Warden, I wasn't hunting deer. I was killing over corn. I don't need no license for that, right?" laugh


Molɔ̀ːn Labé Skýla!