The sense of “live" that you've pulled from Collins is a very specific use. The only example I can think of for that specific sense is in the informal fixed phrase “live wire”:
- (informal) an energetic or enterprising person
- a wire carrying an electric current
Source: Collins Dictionary – definition of “live wire”
Generally speaking, if you want to say someone is “full of life” (unusually energetic), you'll be better off with “lively”. For example:
“He's just this great old guy — he's confined to a wheelchair, but he's so lively. He's ninety-five years old, yet you wouldn't know it to talk to him.”
Source: Jelly's Gold, by David Housewright
If you want to use “live” in its usual sense of specifying whether or not someone or something is dead, pay attention to the first definition of the adjective:
- (prenominal) showing the characteristics of life
Source: Collins Dictionary – definition of “live”
“Prenominal” means that this adjective is used before the subject. Therefore “a live person” is correct, and means that the person is not dead, is not a robot, is not a recorded voice on the phone, or, as in the following example, is not plastic in the shape of a person.
Plastic mannequins are helpful for this kind of work, but the clients usually prefer a “live” fit or showroom model so they can see how the fabric moves with the person, and they are able to ask questions about how an article of clothing Feels. A live person also gives a more genuine feel as to how the fabric is going to drape. Or if an outfit is pulling somewhere that just does not feel right, a person (as opposed to a mannequin) can say something about it.
Source: Break into Modeling for Under $20, by Judy Goss
“An alive person” is not correct, because “alive” is postpositive, i.e. placed after the word that it refers to. That means you would want to say “A person [who is] alive”. This syntax will require you to include a being verb in order to name the state of the person. This usage is usually less metaphorical, and is generally restricted to cases were there are two options: alive or dead. For example:
Burden of proving that [a] person is alive who has not been heard of for seven years.
— Provided that when the question is whether a [person] is alive or dead, and it is proved that [the person] has not been heard of for seven years by those who would naturally have heard of [the person] if [the person] had been alive, the burden of proving that [the person] is alive is shifted to the person who affirms it.
Source: Of the Burden of Proof, §108 in The Law of Evidence by M. Monir