A turn indicator is a rate based gyros ie the faster the heading change the more it reacts. It doesn't react to angle of bank except indirectly when AoB results in a change of heading. Roll the aircraft while maintaining a constant heading (sideslip) shouldn't show a reaction.
An attitude indicator reacts to show angle of bank and pitch attitude. It doesn't react to heading change eg keep the wings level & yaw the aircraft shouldn't show a reaction.
A turn co-ordinator is like a turn indicator but with the gyro's axis inclined w.r.t. the longitudinal axis. This makes it able to sense aircraft roll a little bit. The idea being that AoB usually preceeds a turn so you have an earlier - or perhaps, more obvious - opportunity to recognise a need to make a control input to stop or prevent the heading change sooner.