“Pull in to” or “pull into”? “Head in to”, “Head into”, “head to”, or “head for”?

To pull in indicates a change of direction: “She pulled into the parking lot” -> “She turned off a particular road and into the parking lot”.

OED

7.b. Of a driver or vehicle: to drive off or to the side of the road, esp. in order to stop.

2001 T. Hanley in M. Hickey Irish Days (2004) 125 This side of Tulla there was a bit of a blind laneway and I'd pull in there and have a flask of tea.

To head = to move in a particular (usually forward) direction

OED:

20. Originally Nautical. With adverb or prepositional phrase.

a. intransitive. To move forwards in a specified direction or towards a particular place or thing. Later also more generally: to go, make one's way. Also transitive: to follow (a course, way, etc.) in a specified direction.

2008 A. Nersesian Sacrificial Circumcision of Bronx 261 Once he had located the address of the Russian embassy in the phone book, he headed over in a cab

"To pull in" is an intransitive phrasal verb. You can just say "Pull in!"

"To pull into" is transitive, you cannot just say "Pull into!", you must pull into something, e.g. "Pull into that car park!"