I don't agree with this distinction.
However, there is a different distinction: "to retire" emphasizes the action of retiring whereas "retirement" emphasizes the state of having retired.
Thus, "He prefers retirement" says more about his state of being: he enjoys the gardening he missed out on while working, etc. Whereas "He prefers to retire" means it's the action that matters: "He was given a choice of being fired, which carries some shame, or of retiring, which also provides benefits. He prefers to retire."
Put differently, think about what would happen if you added "over" at the end. "He prefers retirement over ...", eg, "He prefers retirement over working", vs "He prefers to retire over ...", eg, "He prefers to retire over being fired".
I get your point but what we're doing is translating this sentence as closely as possible. Here we have an infinitive following the verb, so it would be a simple and direct translation of "He prefers to take his retirement." I would suspect that Duo would accept that answer; if not, it should. Duo likes "He prefers to retire" which is a reasonable approximation.
It is closer to "to retire", literally "to take his retirement" (to retire) or "to take his pension" (U.K.). The idea is that he is still working but perhaps he will leave his employment in order "to take his retirement". Using "prendre sa retraite" would indicate that he is not already retired. That would be "Il est en retraite" = He is retired.
No, it can only mean retiring as in to take your pension.