Picard is absolutely science. He's interested in the why and how of everything. He doesn't require too much summing up when a Sci officer goes into the gobbldy goook, and most importantly, he's more of a thinker then anything else. He always looks at things from every single angle he can. You must always remember too, that almost every time a tactical situation popped up, he asked for suggestions. Usually from Will or occasionally from Worf, Two of the most tactical officers on the ship.
I was thinking along similar lines. You can almost classify each one by their weaknesses, although that supports my multiclassing hypothesis.
Kirk needed briefing in both science (medicine and exotic science) and engineering. Think how many times he asked Scotty or Spock for advice. He's not an engineer or a scientist. Thus Kirk is a Tac.
Picard generally sought counsel from Worf and Riker (and sometimes Data) in tactical matters. Although he was a competent tactician in some regards, his greatest ship maneuver was rooted in a scientific trick (eg. The Picard maneuver.) I can count on one hand the number of times I vividly remembering him set foot in main engineering and he delegated engineering tasks almost totally. You can be sure he's NOT an engineer. It is less clear that he's not a tactician (he is a war veteran and the movie Picard is almost straight Tactical, taking charge and getting into scrapes) but Science seems most true.
Sisko is an interesting case. His reliance on Dax and Bashir heavily rule out Science. (Do you remember Sisko ever personally conducting research, autopsies, particle scans, etc?) He was not inclined to lead from behind in combat, however, and we saw him design, build by hand, and repair ships in the engineering room. It really depends on if you see managing fleets, ship combat, and wrestling as inherently Tactical but he did defer to Kira and Odo and Worf on security and police procedure. Conclusion: you can't rule out Engineering at all (he tended to work over O'Brien's shoulder -- how many times did O'Brien say that Sisko was breathing down his neck?) and while not competent in all areas of Tactical (police procedure and security detail; he was more of a big picture guy) you can't fully rule out Tactical. The big thing is, he's really not Science.
Janeway is another interesting case. Her background is explicitly Science. However, if we're looking at where she delegates, well... She took FULL responsibility for every aspect of operations and tended to avoid delegating whenever possible, which was played sometimes to neurotic extremes. She spent a good deal of time in Engineering and was not only a research scientist who studied phenomena but someone who knew how machines worked and how to get more out of them. Similar to O'Brien and Sisko, B'Elanna always seemed to regard Janeway as hovering over her. There's also not an explicit science officer aside from The Doctor and Janeway never seemed to defer to anyone else on scientific matters, even overriding the Doctor's opinion. The area where I suppose she delegated the most was Security, leaving these details largely in Tuvok's hands but also taking counsel from Chakotay. She's explicitly Science. You can't say she was disinterested in Engineering (as Kirk and Picard were) either. If you had to eliminate one, it would be Tactical, if (again) you assume Tactical and ship combat prowess are not direct correleries, which they aren't the way this game is setup. Someone who is deadly with exotic ship tricks would probably be less likely to be Tactical if anything and the real clue is that a Tac's ship should be getting heavily banged up in every battle, as they play closer to the edge and have less prowess in keeping the ship unscathed.
Archer... I'm fuzzier on. But my feeling is that he wouldn't have needed MACOs or been pushed to do some of the things he did as a Tac, nor would he have been in competition with T'Pol if he were totally a Scientist or Tactician. I think we saw less of him delegating specific tasks overtly so it's harder to judge him by what he didn't do but through T'Pol and the MACOs, delegation was forced on him because Science and Tactical were areas where he was considered unproven, especially Tactical. I'd peg him as definitely not a Tac (in spite of whatever innate traits he had like physical strength to go up against a Vulvan) and maybe not well regarded as a Scientist. That leaves engineer with Science up for some debate mixed in there but Tac being the area where, however good at it he may have come across, nobody trusted his judgement at first.
Heh, nah it looks this confusing only because Brannon/Braga had no fricken idea how to develop that char properly.
Sci definitely for Janeway, the time travel and Leonardo da Vinci holodeck eps show that clearly.
Also, the show's bible says she was a science officer on the U.S.S. Albatani before making captain.
And they did highlight her scientific knowledge in the early episodes.