Are you glossing another language? If those examples are English, the first one is not in any person (since it has no verb to agree with), and the second and third are both unambiguously third person. A word does not become `second person' simply because it refers to the person you are talking to.
Haha. Then we should really say: "Mother am busy right now"!
Eh, what? "am" is the first person singular copula. "Mom" is third person.
I am busy — (no equivalent)
you are busy — (no equivalent)
s/he/it is busy — mom is busy
we are busy — us moms are busy
you are busy — you moms are busy
they are busy — moms are busy
Third person.
It's a joke! Of course you don't say: "Mom am busy"! The point is, the grammatical person and the actual person seems to be different here! For example, if I were the mother, and were talking to my child, I'm a first person, as I'm talking about myself:
Mom needs some time alone right now. (I'm talking about myself)
In second person we prefer using vocative instead. However as vocative always addresses the listener it's actual person is second person:
Mom, I want to play! (The child is referring to me. I'm the listener.)
So that leaves the actual third person to be in every case we refer to someone who isn't us and who isn't the person(s) we are talking to.
But dad always plays with me! (but now the child is referring to someone else)
The grammar I've showed you here is only available for two nouns "mom" and "dad", and names and titles. But then we don't refer to ourselves by our name do we? Like this:
Vir likes grammar. (I'm refering to myself)
But in Japan, it would seem that female's may actually do this because they think it's cute.
So if we had these distinctions we might actually conjugate the verbs according to this. That was what I was getting at. One thing I believe will never change though is the usage of the third person to speak in an indirect manner when you are trying to hide your own personality:
I heard someone was creating a language.
He wouldn't be go by the name Vir Docilis at the forum would he? (I'm speaking)
Here the speaker wants to pretend that the person he is talking about is not himself so he's using third person. As I'm saying, I'm not talking about the grammatical person here, but rather the actual person. But even if we look at this from the grammatical person perpective: "What happens when you change the nouns in the examples to pronouns?". The answer is, that they change into the persons I'm talking about. For example:
Mom needs some time alone right now. -> I need some time alone right now.
My conclusion is that the grammatical persons in our languages are irregular. I'm putting suffixes after the nouns in my language, making this completely regular, meaning that when you change the nouns into pronouns, they will be similar to the pronouns. Like this:
Mom(I) needs some time alone right now. -> I need some time alone right now.