There are lots of ways of talking about starting cars. Except for (1) and (2), these are fine.
There is an English grammar rule being violated in (1) and (2).
Native speakers know it, because they follow it, but they usually can't state it.
Non-native speakers need to be taught the rule, however, because it's not obvious.
- When a noun modifier consists of more than one word, it goes after the noun it modifies.
When a noun modifier consists of only one word, it goes before the noun it modifies.
Mnemonic: an eleven-year old boy versus a boy eleven years old
Since turned off/on is more than one word, it must follow car, not precede it.
A car turned on/off is OK, but not *A turned off car.
(There is a single-word metaphoric term turned-on/off, which refers to affectual display,
- He describes himself as a turned-on Christian.
but it's only metaphoric, and doesn't refer to actual machine power status.)
This rule is the one that makes the title of the famous cat video
so odd-sounding to native English speakers.