A PA34 pilot misinterpreted his clearance out of CRQ and turned on course prematurely.

Narrative

Aircraft X was provided the following clearance from CRQ Ground: 'Cleared to ZZZ; fly heading 25-degrees vector; V23; SLI; V459; LHS; Gorman; V137; AVE; then as filed; maintain 3;000; expect 12;000 in 10 minutes; departure frequency XXX.X; squawk will be assigned before takeoff'. I read back the clearance exactly as stated. The Standard Departure procedure from CRQ for Runway 24 is; 'Climb runway heading (240-degrees) to 2800. All aircraft climb on course'. Two days prior; I was given a departure of 'after takeoff; turn to 250-degrees; so I was familiar with the departure procedure which differed from the Standard. Knowing I was to fly V23; I set V23 on the aircraft GPS and VOR by setting the 326-degree radial from MZB and 145-degree radial from OCN. Upon departure; I flew the runway heading until the beach (as per CRQ guidelines); then turned to a heading of 250-degrees While in the climb; but before reaching 3;000 feet; I was instructed to climb to 8;000 feet. I received no further instructions but two frequency changes; each one to SoCal Approach. A track of 250-degrees intercepts V23 and; upon reaching V23; I turned on-course; as cleared. Because the clearance deviated from the Standard Departure Procedure; and I was given a radar vector of 250-degrees to fly (which differed from the 'climb runway heading...climb on course' as set forth in SW-3); I believed the 'vector' within the clearance was this 250-degree heading. As such; I believed once I reached V23; I was cleared to fly V23. At that point; SoCal Approach; instructed me to fly a heading of 270-degrees which I did. A hand off was made shortly thereafter (to a male controller); who simply gave me a phone number to call upon landing and informed me I had made a 'pilot deviation'. I was provided no further explanation but was instructed to fly direct 'SLI'; which was the next waypoint in the original clearance and a waypoint on V23. Upon landing at ZZZ; and calling; then receiving a return call; I was told there was a 'loss of separation' between my aircraft and a corporate jet; which departed CRQ after me. I was told my clearance differed from what I was told and Mr. X did not know about the 250-heading vector assigned by CRQ Tower. I believe the original clearance was followed and the 'vector' I was instructed to fly was the 250-degree heading; since it differed from the Standard Departure Procedure for CRQ. Once I approached V23 on the 250-degree heading; I believed it proper to fly V23 since I had completed the 'vector' portion of the clearance. Also; if radio communication had been lost; it would have been proper to fly heading 250-degrees to V23 to SLI; and so forth. If there was any confusion; it was partially because the clearance differed from the Standard Departure Procedure and the fact the 25-degree heading was believed to be the 'vector' indicated within the clearance to take me to V23. It was easily confused. In the future; as what happened throughout the remainder of this flight; these issues and confusions will be clarified with much more specificity (including requests for specific courses; intercept courses; and navigational aid tracking); especially when the phrase 'vector(s)' are used immediately preceding or subsequent to the phrase 'Vector-??'. In this instance; the 250-degree heading instructed by the CRQ Tower was easily confused and was reasonably interpreted as the 'vector' contained within the initial clearance; since it differed substantially from the Standard Departure Procedures.