I am a "slow" gamer. I enjoy building up a plan over hours and seeing how it plays out. It is also fun to see one's carefully constructed plans layed to waste.
When I'm enjoying my planning, another player can be engaging on a purely tactical level (turn by turn, but not "big picture" planning). This "dual-level" is part of what makes Le Havre so good.
I recently played Le Havre with a gamer who was neither tactical nor strategic. He didn't enjoy the game and made moves which hurt his own position. I felt bad because he had wasted 3 hours on a game he didn't enjoy.
We should have just quit as soon as he realized he didn't like the game, but we were stupid.
Later on, when we were playing Dominion (a game he enjoyed a lot more), I think I started to understand what he enjoys about games and how Le Havre wasn't satisfying for him. He played very strongly, controlling the tempo and pacing of Dominion, while I was seeding my deck with synergistic cards as part of my long term plans. We ended up with a tie game. And we had so much fun playing that the ending didn't matter.
I guess the main thing I took from this isn't a third category (to go along with tactic and strategy) but the idea that categories never tell the whole story. Just play games that are fun for you and don't worry about labels.