Runway heading is a heading, not a track. So no drift correction should be applied.

The French radiophraseology manual, which I assume is based on EASA/ICAO, does make the distinction between the two. So in one case you should fly runway HEADING, and in the other you should track the extended centerline.

Contrary to the US, in Europe, or at least in France, I am usually instructed to fly "runway track" instead of "runway heading". On my last flight however, I was for the first time instructed to fly "runway heading"...

Dear colleagues

Just think what the ATCO knows about the changing wind and changing airspeed during the climb?
For this he would need a dynamic crosswind calculator and would have to constantly advice for new headings.
Most of you have GPS data showing the track, or you use landmarks, which you need, if you want to fly the correct inbound- and outbound-routes.
The controller thinks geographically.
So he certainly means the track not the heading, or do you ask him the heading for a crosswind landing?
Try to go for a week as an ATCO assistant, this would be a very interesting week.
Conclusion:
ATCO information refers to course over ground, either with landmarks, radio-fix or track.

As a retired controller with more than 30 years in tower and approach experience I dare to state that most controllers have little knowledge on flying aircraft. Technically "runway heading" is a heading and drift should be disregarded, however I would compensate a simply fly "runway track". Of course, a pilot can always check with the ATCO or readback "cleared for take-off, fly runway track" and wait for the reaction of the ATCO.

The ATC usually gives heading instead of track. It is because the older planes has no TRACK maintaining capacity like the new ones, and they do not calculate with wind. So RUNWAY HEADING means heading maintaining instruction. Some times they already use TRACK, but if you can not maintain the desired track, you can just say: unable.
So Hermann is not right, Paul is. The standard English radio communication phraseology ensure this also.

Of course then you could get into questions of whether we're talking about magnetic or true heading. Assuming it's magnetic and you'd be flying to the precise degree, not the nearest ten degrees, as denoted by the runway designator. You're flying VFR so you wouldn't even be expected to fly accurate headings, and you're unlikely to be flying often in conditions that would be blowing you significantly off-course at such low altitude. It's worth a moment of consideration but only because it can broaden your awareness of all the other factors in play (inter alia those that I have mentioned here and that other contributors have mentioned above) and to hopefully illustrate which are the most important/relevant when flying an aircraft. (My point here is to show that your doubt about the exact heading to fly should be the least of your concerns).

What would be at least as interesting to know would be WHY you are being asked to do it. Is it for noise abatement? In which case, just look out at where the populated areas are. Is it because of other traffic? Try to make visual contact. Etc....

Let's not make a simple task difficult. "Fly RWY HDG" means just this: continue to fly the RWY HDG you are on. If ATC isn't happy with where this is leading you, they'll just say "ABC, turn right/ left xxx°"

This is pretty straightforward: runway heading is what it is. ATCO will consider affect of the wind if any. ATCO has in many major airports rather good information about the wind at different altitudes.
Heading is used in radar ATC service. Nowadays it could be possible to use tracks (technically), but You know how easy it is to change something in aviation globally (like NOTAMS). Takes decades.

I would dare to say, when asked to fly RWY HDG, it means fly RWY QFU, once airborne, in order to maintain the RWY QFU, you need to compensate for the wind, so you will be flying a track.
I think, what ATCO have in mind, is that you maintain a straight forward flight from the extended center line, if you don't correct for the wind, you wouldn´t be able to do that.
Have safe landings

The answer is in Part-SERA (download the Easy Access Rules pdf, see page 150):
Both heading and track are possible and standard phraseology:
- When heading is to be followed the instruction is: CONTINUE RUNWAY HEADING.
- When a specific track is to be followed: TRACK EXTENDED CENTRE LINE.

So, when instructed to "continue runway heading", you do no correct for drift.

Hey all,

thanks for all the answers, this was overwhelming!
I got a request from ATC to 'prepare to fly runway heading 1000ft' (there was traffic in the pattern) but later that changed in a 360...
Later at home that made me think 'what did she really mean by that?', so what better place to ask here!

Hope to see at least some of you on the 11th!

@Bas, thanks for the page number of The Easy Access Rules, that was very helpful), I could not find using my google-fu...