Runway Heading
I quite agree there is a lot of confusion on this one.
If you look in the phraseology section of MATS Part 1, there is actually no approved standard phrase 'fly runway heading'
If an ATCO wants you to go straight ahead (which is usually what they are after in these circumstances) then the approved phrase is 'after departure climb straight ahead' or 'track extended centreline'. IMHO you should never hear the phrase 'fly runway heading'. If an ATCO wants you to actually fly the runway heading he should say 'after departure climb heading xxx degrees'
I have often given the instruction 'climb straight ahead' to have it read back as 'fly runway heading' - and have also been caught out when I have expected the aircraft to climb straight ahead and it has gone veering off in all directions!!