“Into” implies entering or being a part of something. It is it’s own preposition. “I’m into cars.” “I’m going into the store.”
“In to” are just two words that sometimes end up next to each other. “I’m going in to use the bathroom.”
I was taught once that when using “in to” the sentance should still make sense if you stop after the word “in.” “He went in to buy some coffee” still makes sense as “He went in.”
Correct (also correct comma usage)
“One day leading into the next.”
“I put my fist into her box.”
“I put my fist in, to teach her a lesson.” (this doesn’t necessarily need the comma, but is gramatically correct either way, depending on the way the writer wishes it to be read.)
“I pushed my car into the garage.”
“I pushed my car in, to work on it out of the rain.”
“Additionally, I believe commas are being used way too liberally, especially by those who are under the impression they have a good grasp on grammar.”