From usingenglish.com...
The finite forms of a verb are the forms where the verb shows tense, person or singular plural.
Non-finite verb forms have no person, tense or number.
finite I go, she goes, he went.
non-finite To go, going (the infinitive, and the gerund/present participle).
Thus in OP's first three examples, coughs, had compromised, will have gone are "finite" forms of the verbs to cough, to compromise, to go. In the second and third cases, the modal auxiliary verbs will and had/have are the only ways in English to indicate verb tense, aspect, and modality, apart from the endings -s (present tense, third person singular), -ed (past tense), and -ing ("continuous" aspect).
In OP's last example, were is a "tensed" (finite) form indicating past action and plural subject (or second person singular "you"). So those verb forms change for present activity or singular subject, but the "non-finite" present participle talking doesn't...
The girls were talking to John. (past action, plural subject)
The girls are talking to us. (present action, plural subject)
John was talking to me. (past action, singular subject)
John is talking to you. (present action, singular subject)
Also note that it makes no difference whether the object (John, us, me, you) is singular or plural.
"From now on" requires present or future tense, as it refers to something that starts now and continues.
If grandpa Peter tells that story to his grandchildren, he could say "From that day on, I have been suspended from school".
Best Answer
You are right with your assumption:
The word 'the' is a definite article. In this case, using 'the' before the noun 'people' makes it (the noun) unique. It makes clear that those people (and only those people) have something in common: they were present when the accident happened.
Further reading: https://www.ef.com/english-resources/english-grammar/definite-article/