Hey Jakob_Knight,
I find it interesting that you separate "good player" and "good teammate", why can't these be mutual? I consider myself a good player as well as a good teammate and try to do what is best for the team in most cases. Sometimes this is tough to do because the battle develops in a certain way, but for the most part doing what is best for the team is also best for you. This all boils down to what you have set as a goal personally, correct? I like to win so I want to do what is best for the team and I want my contribution to help the team win.
Signs of a good player (experienced player) IMO:
Target Prioritization: What they shoot and when they shoot it is a good sign for you to tell if a player understands their role.
If in a DD, they pay attention to the other team's lineup and they identify where the radar is before moving onto a cap. If they do move onto a cap the DD doesn't sit broadside and always has a way out.
Early map movement: Understanding that you cannot just give up half the map and focus on 2 caps only. Yes, you can move the bulk of your ships away but you still need a deterrent/harassment group so that the enemy team doesn't just sail around and flank you.
Knowing where you should go with your ship based on strengths and weaknesses.
Doing the little things: pointing out where a DD was (not spamming), focusing down lower health targets and overall putting yourself in the best spot to help your team win.
These are only a few things that I can think of off the top of my head.