You would be right if the turn indicator/coordinator was meant to show the same information as the AI, but that is not the case.
The turn indicator (which is the single arm pointing upwards, swinging to either side) is really telling you that the aircraft is turning, not that it is banking.
The turn coordinator (which is the model aircraft seen from behind) is also intended to show you that the aircraft is turning, but here the axis is tilted off the vertical so that it indicates change of bank as well.
The slip indicator is just the ball and is driven by gravity/centripetal force...
Because the turn indicator/coordinator are based on the turn of the aircraft, they then allow to be calibrated to show a rate 1 turn - something you could not do based on bank alone without some complex mechanism involving airspeed etc.
The coordinator is usually preferred to the indicator because it does show change of bank before the actual turn has started, giving you a bit advance warning. However, it will overindicate, e.g. it will show that you are doing a rate 1 turn before you actually are doing one. Once you have a steady bank/turn, it accurately gives you your rate of turn again.
Of course, neither gives any pitch information.
The AI does not actually show that the aircraft is turning, it just shows whether it is banking or pitching.
If you put the aircraft into a steady sideslip, the AI should show a banked aircraft, but a turn indicator should, once the aircraft is in a steady slip, not show anything at all (since you are not turning), whereas the turn coordinator will also, after showing the initial setup of the slip, return to 'flat' since it shows the change of bank but not the actual bank.
Of course, the slip indicator/ball will be way off to one side...
B.