Air Carrier Captain reported confusion while flying the VICKI9 STAR into KOA. The Captain asked ATC for their interpretation of the arrival but they were unable to supply an answer. The Captain maintains the transition is unclear regarding which routing to fly.

Narrative

There is some confusion on the arrival/approach into KOA due to some inconsistencies in charting and in ATC instructions. Take a look at the VECKI9 arrival and the ILS17 approach. There are multiple places to join the arrival and multiple approach transitions. Our flights from the mainland use two of these.The first one is arriving over UPP VOR. In this case; the routing from UPP to VECKI as shown on the arrival is exactly the same as the routing from UPP to VECKI as shown on the approach transition. So; in this case when you're still well north of UPP and ATC says 'cross UPP at or above XXX; cleared ILS 17 approach;' there is no confusion. It doesn't matter whether you fly the arrival to VECKI and then the approach or if you fly the approach transition from UPP because it's exactly the same.However; the second way we get into this airport is by joining the arrival at ONOHI. In this case; the routing from ONOHI to VECKI as shown on the arrival is *different* that the routing from ONOHI to VECKI as shown on the approach transition. It's not much different; but the routings diverge at the TAMMI intersection. The arrival goes straight from TAMMI to VECKI (it's actually depicted as a heading on the chart; but the FMC is coded direct). The approach transition goes from TAMMI to QEJXO (also depicted as a heading on the chart; but coded direct in the FMC) to BAYCA to VECKI.So if you are north of the ONOHI intersection and ATC says 'cross ONOHI at or above XXX; cleared ILS 17 approach;' which of these two routes do they expect you to fly? The arrival to VECKI and then the ILS with no transition? Or the arrival to TAMMI and then ILS with the TAMMI transition?I've actually flown it both ways; and ATC doesn't seem to care. I specifically asked them on this flight; and they seemed confused by the question.So how do you solve this problem? It could be fixed if they just aligned the routes on the arrival and the approach transition so they are the same when coming from ONOHI; just as they are when coming from UPP. Alternatively; ATC could be more specific in the clearance. Maybe something like 'after TAMMI; cleared ILS 17.' That makes it clear that you are supposed to fly the approach transition after TAMMI and not the arrival all the way to VECKI and then the approach.At most airports; the STAR ends where the approach transition begins; which also prevents this from being an issue.Please reach out to Honolulu Center.