This may be a UK/CAA phraseology quirk, but I’ve heard controllers use the phrase “fly heading 090” and “turn right/left, heading 090” seemingly interchangeably.
When would a controller use each phrasing?
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In general, I use "fly heading" when I'm not 100% certain of an aircraft's heading (e.g. if they're proceeding direct to an airport fix) and there's a chance they might be on the "wrong" side of the heading.
For example, imagine that an aircraft is on a ground track of 290°. I would like to tell them to fly a specific ground track, but the rules don't allow that. I want them to fly heading 300° for now, but I don't know what the winds aloft are doing—they might be on a 280° heading to get their 290° ground track, or they might be 315°. Instead of telling them to turn in a specific direction, I just say "fly heading 300" and let them turn in the shortest direction.
Most of the time, though, I specify turn: left or right. And if it will be the long way around I repeat myself for emphasis, or maybe even say "turn the long way" in so many words.
In the U.S. the example you show in your question would be correct (standard) ATC phraseology.
These two types of phraseology are often used interchangeably, but there are circumstances that the use of one method as opposed to the other would be appropriate.
For example, if you were heading 180 degrees and the controller wanted you to turn to a heading of 360 degrees, he/she may want you to turn to this heading in a specific direction (i.e., "turn right heading 360" as opposed to "turn left heading 360").
It's also, in my experience, far more common to use this direction of turn method, as opposed to the fly heading method in the course of issuing radar vectors (with respect to the two methods you are asking about in your question).
If you were instructed to Fly Heading (degrees) then the pilot would be "...expected to turn in the shorter direction to the heading unless otherwise instructed by ATC." This information is noted in the Pilot/Controller Glossary listed under "Fly Heading (degrees)" located in Appendix PCG-1 of the FAA's Aeronautical Information Manual (AIM), linked below.
An example of when ATC might use the fly heading method would be if an aircraft was on an arrival procedure/transition (e.g. a STAR) and was on a course between fixes of 080 degrees and the controller wanted to take the aircraft off of the arrival procedure prior to the next fix. In this case the controller could say "fly heading 080 vector to the ILS runway 03." This is just one example, but there are other situations the fly heading method might be used.
Here are the appropriate sections from the FAA's Order JO 7110.65AA, paragraph 5-6-2 (Radar Vectoring Methods) and the noted excerpt from the Pilot/Controller Glossary located in Appendix PCG-1 of the FAA's Aeronautical Information Manual (AIM):
(highlighting is mine)
(highlighting is mine)