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I believe the definition being used here would be along the lines of “to become infected or affected by,” “To come to have gradually; acquire,” or “To become affected with; contract.”
The meaning is then “the pugilist [is affected by] nonlethal damage only,” which is roughly equivalent to “the pugilist isn't affected by damage that isn't nonlethal.”
The problem with that interpretation, aside from its obscurity, is that the 'is affected by' is actually 'is affected by [a disease; a contagion]', and is supported by the example "He developed pneumonia after a bout of flu."- the synonyms are also supporting- stricken, came down with.
A person comes down with the flu, or is stricken with pox. They're not stricken with a bad case of the knife wounds, or come down with fifty degree burns.
Non-lethal damage isn't a disease or a contaigon- or an ongoing deabilitating process.
Moreover, taken each and every possible definition as a whole, all but the singular, misused definition of 'is affected by' support that the meaning is not 'he is immune to lethal damage'.
EDIT: You've also, in your own quote, modified the RAW. The example you're using is "To become affected with;", but then in your quote you alter it to "To become affected by"
So it's not 'I've found a definition that makes this feature make you immune to lethal damage', but I've found that a definition of a word used in a defintion for a word used in this feature allows me to make this feature make you immune to lethal damage'- at this point we're two steps removed from the feature itself.
So yeah, it doesn't mean you become immune to lethal damage. It doesn't *say* you become immune to lethal damage even under a strict reading. It only says that by your homebrew alteration to the definition of the definition of the word used in the feature.