Very good @Snowbird, I was just searching for that very passage and yet it eluded me. 
The rule mentioned above under Mv.VII.22 is given in specific reference to excess robe cloth, so Thanissaro uses the great standards here to apply it to other items. He uses the word"leftovers", which makes it sound as if he is talking about food, but I think, in the context of the passage this appears in, he is actually referring to other leftover requisites, such as robes and bowls.
Leftover food (as your monk friend monk said) would be simply classified as relinquished by the monastic once they have finished with it, such as at a monastery buffet table, the food no longer belongs to the bhikkhus/bhikkhunis once the Sangha has taken their food, and then it belongs to the novices and anagarikas and lay people who follow.
However, I think it feels like a different situation if a monastic is giving food away before making use of it, or taking something out of their bowl to give to a particular layperson, or distributing other sangha-dana as treats to a certain laypeople. It looks like favouritism, or an opportunity to ingratiate oneself for a future reason, or possibly an exchange or trade. I recall an occasion where I thought I was being really lovely and kind by giving lay people some chocolates that I had excess of, and a senior monk admonished me about that, saying we shouldn’t give out “treats” in this way, especially in public, when the donor might see! I didn’t like being told not to give, which I saw as a positive opportunity to practice a beautiful aspect of Buddhism… but thought about it later on and saw how it could look to others.
So then, imagine my surprise when I found myself telling a novice this very point a few years later, when he did exactly the same thing (
) ! Perhaps, as he had only recently ordained, he was still in giving mode rather than recipient-field-of-merit mode – there is a shift to get used to, moving from lay life to being a monastic – but oh, he was making such a show of being the big benevolent bestower of largesse (albeit with something given to him for free by someone else…) and now I saw quite clearly that it certainly didn’t look good; was he playing favourites? what about the people nearby who missed out? What if the donor saw him being generous with things they’d given!? Did he just want to look super generous to others, out of ego?
This is using gain for gain… I think that’s the reason there’s a culture in monasteries of not giving food or medicines to laypeople, and instead relinquishing excess requisites to the store monk/nun for impartial disbursement, or to the anagarikas, who as monastery attendants can take what they need and then disperse anything remaining (usually to charity).