When someone says “Thank you”, we may say “That's all right.” I wonder, what does “that” refer to ? What's the meaning of “all right”? When you agree someone's proposal, you would say “OK” or “All right.” What are these sentences short for?
Consider this exchange:
Speaker 1: Thank you. Speaker 2: That's all right!
In this case, speaker 2 is very casually acknowledging that speaker 1 has thanked them. It is somewhat lighthearted. This is similar to, but less polite, than an actual "welcome":
Speaker 1: Thank you. Speaker 2: Welcome!
These are both very informal and and would not be used in a professional or business setting, or where you do not know the speaker well. It is considered much more polite to say "you're welcome" when you are thanked.
In the second case:
Speaker 1: Thank you. Speaker 2: All right.
I would not say this, it definitely comes across as rude. In this instance, it is only an acknowledgement that speaker 1 has spoken, without speaker 2 offering any sort of welcome in return. It would be the equivalent of:
Speaker 1: Thank you. Speaker 2: Ok.
It's always better to be polite in formal situations. I always say "you're welcome" to everyone.
Anyone: Thank you! Me: You're welcome!
To answer the second half of your question:
When you agree someone's proposal, you would say "OK" or "All right." What is this sentence short for?
Answering someone's proposal with "OK" or "All right" generally means two things:
1) You have confirmed you understand what they are talking about.
2) You are agreeing with their proposal.
It is a colloquial shorthand for saying "I understand and I agree."
Up to my knowledge both "All right" and "That's all right" are used in casual conversation. But in formal conversation, mostly "All right". Both have the same meaning, no difference in meaning. I do not know beyond this. I'm not a native speaker.