Is there a specific verb or phrase for holding or carrying something in this particular way:
Is it just "to hold or carry [something] under one's arm" (as translated literally from my mother tongue) or is there something more idiomatic?
"He held the chicken tucked under his arm." and "He tucked the baguette under his arm." I would say they don't sound as accurate as they actually are, because people are not as likely to refer to their armpits, but these examples describe (to me) exactly what is happening in your images.
This is called 'carrying in the crook of your arm'.
One of them was cradling a sub-machine-gun in the crook of his arm.
Carey stood up, the fish held in the crook of his arm, as you would hold an infant.
Mortified, she turned her face and hid it in the crook of her arm.
The Ngram shows the expression to date from about 1898 so it would appear not to be derived from the shepherd's crook.
Ngrams show that it is prevalent in both AmE and BrE from that date.
The earliest citation I have found (to date) is its use by Sir Arthur Conan Doyle in one of his earlier novels (not Sherlock Holmes) which would have been written in about 1890 as this was his 'early' period before he wrote the Holmes novels.
Eytymology of the word 'crook' :-
Old Norse krāka hook
Middle English crok(e) 1125–75