Flight Crew Air Transport Pilot (ATP)
Confusion; Distraction; Situational Awareness
ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Provided Assistance; Air Traffic Control Issued New Clearance; Flight Crew Returned To Clearance
Human Factors; Procedure
Assigned the MIA Hedly 1 Departure from 8L.  Clearance was misunderstood to mean we were to fly the departure as depicted.  We reviewed the departure and the flight plan IAW the FOM and AOM procedures.  Upon reaching 8L we were cleared for takeoff.  I do not recall hearing an assigned heading nor were we told to 'RNAV to Senoy'; the first fix on the DP (Departure Plate).  We departed in Nav mode.After takeoff the FMS captured and steered us essentially straight ahead to Senoy.  At Senoy; we began a left turn toward the next fix Cimba.  At that point ATC queried and suggested that we may have taken someone else's heading.  The Second in Command (pilot not flying) replied that we were following the SID as cleared.  ATC said that was incorrect and gave us an amended heading; while I kept the aircraft slow until we sorted it out.Miami said they would talk to the tower to figure out what we were assigned; but we never learned the result of that call; if it was ever made.  The most telling comment from ATC was 'You wouldn't be the first one to do that; it's something we're working with the FAA on right now.'  That was the last thing we heard about it.It seems that there was some confusion on the part of the Second in Command when the clearance was received.  I was not present at that time; but I found it interesting that even following the FOM procedures for route verification; that initial copying of the clearance is a potential single point of failure.It seems that there is some known issue either with this departure or in how it is verbalized by Miami ATC. The controller's statement suggests that this is not an isolated incident; but it is currently being addressed.  I am not suggesting that we require both pilots to be present for the copying of the clearance. There are far too many duties placed on us already; and more show up by the day; including now passenger iPad maintenance; etc.There is simply too much required by our Company in getting a plane ready for flight to have yet another requirement for both pilots to be in the cockpit to receive a clearance.  Errors will happen; and I will be more cognizant to ask a question if I think something is out of the ordinary.  Adding another layer of redundancy in this case will likely lead to something else; perhaps outside of the aircraft; to be missed; unintended consequences.
I called Miami clearance delivery to get our clearance from MIA.  According to our company filed flight plan; we were filed for 'MIA..HEDLY J53…' I copied and read back 'Aircraft X cleared via radar vectors HEDLY as filed; maintain 5000 expect 400 10 min. after departure. Departure on 119.45; squawk XXXX'. The controller on clearance delivery corrected me by saying 'Vectors to HEDLY1 RNAV departure.'  With his emphasis on the HEDLY1 RNAV departure; which actually ends at HEDLY; I misunderstood the clearance to think that we were to takeoff and fly to SENOY and then follow the HEDLY1 RNAV departure. Upon reviewing our clearance with the captain prior to start; I was so certain that we were to fly the RNAV; that the captain didn't have any reason to question the clearance. We took off on runway 08L and flew out to SENOY where we commenced a left turn towards CIMBA on the RNAV departure.  Shortly after that turn; our departure controller said we must have taken someone else's turn by mistake.  I said we were just flying the RNAV departure.  We were given a tight turn to west.  I apologized for my mistake and we received a comment from the controller that we were not the first ones to have made that mistake and that we're working with the FAA right now to fix that.  Very shortly after that we were cleared direct to HEDLY and as filed.I was very mistaken to think that a vector meant to fly the published departure; but I was confused by the correction to my read back that it was the HEDLY1 RNAV departure not just vectors to HEDLY.  While it is not required; the tower controller who cleared us for takeoff might have said 'Fly runway heading' or assigned us a heading for vectors; though I am not certain that it would have alerted me to the error in my thinking of the RNAV departure.On our next leg flying out of Airport Y; we were waiting for 6 to 8 other departures prior to ours.  Airport Y was using the RNAV departure; and every single aircraft departing on RWY XY was 'Cleared for takeoff runway XY; RNAV to [fix on SID]'. That left no doubt as to what that crew was expected to do.
Pilot and co-pilot report of confusion on departure of flying the RNAV departure or flying to a fix.  ATC corrects them and advises that they aren't the first ones to do that.