For what it's worth, US usage of the term is now made explicit in the Pilot/Controller glossary:
<font face="Verdana, Arial, Helvetica" size="2">RUNWAY HEADING- The magnetic direction that corresponds with the runway centerline extended, not the painted runway number. When cleared to "fly or maintain runway heading," pilots are expected to fly or maintain the heading that corresponds with the extended centerline of the departure runway. Drift correction shall not be applied; e.g., Runway 4, actual magnetic heading of the runway centerline 044, fly 044.</font>
It's clear from the above that it does
not mean the same as "climb straight ahead". Controllers choose your words carefully; pilots listen carefully!