What difference in meaning or otherwise does the inclusion or omission of the definite article make in this sentence?
When that day comes, you will be (the) first to know.
I would like to add that in Ngram, the results are pretty comparable.
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Both can be used in a casual context. E.g"You'll be (the) first to know"
Use "The first to know." in a formal context. E.g"You will be the first to know."
I think this is why it is always better to use the correct grammatical from. It enables the reader to carry on without stopping to consider alternative meanings, and perhaps picking up a red herring. Normally I would assume that 'the first' means that you will be told first, either by the speaker or someone else. just 'first' may mean that you will know by some other means than being told, and perhaps will know even before the speaker.
If you said to me, "When that day comes, you will be first to know." I would understand it to mean when the day comes I will be the first one you tell. i.e you will know the day has come and the first person you tell will be me. Whereas, if you said, "When the day comes, you will be the first to know." I would take to mean I will know the day has come before anybody else, including you.
The sentence "You'll be the first to know." has more of a "You'll be the one and only person to know." connotation, but in most cases, you can use them interchangeably.
'you will be first to know' sounds more natural as part of speech and is what I would use when talking to someone, but 'you will be the first to know' is probably more grammatically correct.