That first post indeed says they abandoned the hindrances, but not that they attained jhana. You can abandon the hindrances without entering jhāna, because there is another thing you have to do, which is to become fully separated from kāmā (sensory experiences)". In fact, that this sutta specifically says “while walking” one abandons the hindrances but does NOT mentions jhāna or abandoning sensory experiences, I think speaks in favor of the non-bodily interpretation of jhana. I’d say it purposefully leaves “separated form sensory experiences” out, it and only mentions abandoning the hindrances, the latter of which I agree one can do while walking.
Bhante Sujato has stated samādhi has a broader meaning than jhānas as well, see Swift Pair of Messengers. So has Ānalayo, and I think most everybody, actually.
No, unless you believe one can also think in the fourth jhāna.
(Edit: what I said here before wasn’t right.)
Another text which uses very similar Pali is DN16: “‘Oh, how incredible, how amazing! Those who have gone forth remain in such peaceful meditations (vihārena viharanti).” It’s clear does not mean to say that they dwell in these meditations RIGHT NOW, but that they do so generally. The same is the case in MN122.