What's the word for statements like “Stay with me or I'm breaking your car” It's not blackmail, exactly, so what is the word to describe this type of 'if you do this, I'll do that' scenario/statement?
This is extortion.
The crime of extortion is very similar to blackmail. It also involves obtaining something from another person (usually money, property, or services) through the use of threats of physical harm or coercion.
Generally speaking, an act can be defined as extortion when the actions that are threatened in the unwarranted demand would include physical violence or damage, or constitute an offence themselves. A protection racket, for example (a criminal system where “protection” from harm is offered for money) falls under this category.
The definitions I've found in freely available online dictionaries do not explicitly mention 'services' (compliant behaviour in general; acquiescence).
As DjinTonic has already pointed out in a comment under one of the other answers, the most obvious word for what is described in the question is threat. If this obvious word is not the right one for the OP's purposes, the OP needs to expand the question and explain what circumstances call for a different word.
It is unfortunate that a moderator's locking the comments has made it necessary to post this as a wiki-answer, which makes this page excessively elaborate for a question that arguably should have been closed as too trivial and then deleted (if it turned out that threat was satisfactory for the OP's purposes), or could have been edited in response to comments posted under the question to make the OP's purposes clearer (if the OP had in mind something for which threat is inadequate).