Which is the correct word to use when talking about a person?

"I know a person who has red hair."
"I know a person that has red hair."

"She's the one who stole the jewel."
"She's the one that stole the jewel."

"He is talking to the woman who sews tablecloths."
"He is talking to the woman that sews tablecloths."

"The one who painted the house is standing over there."
"The one that painted the house is standing over there."

Hopefully four examples will suffice.
I'm just wondering whether one or the other is incorrect or inadvisable, when this rule applies, et cetera.

"The person that does X" is slightly less correct than "the person who does X." I suspect I am one of few people who would ever notice, though. It's sort of in the same category as calling a baby "it."

If I remember correctly, and it's been awhile, "who" refers to people and "that" refers to objects and animals and the like. Hope this helps.

My general thought is that it matters in context and which sounds better in the sentence/paragraph...

Well, technically, both are correct...using "who" is slightly more proper, but using "that" is actually slightly more common (at least where I live).

Both "who" and "that" are correlatives, which are a specially type of pronoun (those who disagree should prepare themselves to be smacked with the full force of my linguistic knowledge, such as it is). Technically both of the pronouns can refer to people, but "who" is specific to people while "that" is specific to certain types of grammatical sequences. Sometimes they overlap into areas where both can be used. That is why both are accurate, but "who" is more proper.

So, Sam was right.