When we say generally the ‘shareholders’ have the power to...

When we say generally the ‘shareholders’ have the power to appoint directors, we usually mean a simple majority of the shareholders have this power, i.e. more than 50%. This means that if you hold more than 50% of the voting shares, you have the power to appoint all the directors, and to sack all of the directors.
Under this standard scenario, if you hold say 30% or 40%, you technically have no right to representation on the board. That is, unless the majority shareholders specially agree otherwise.

If you hold a significant stake in a large, widely held company, then you may be able to lobby for a directorship with something less than 50%, say 10-20%, but this is not guaranteed, as Solomon Lew found out in his bid for a board seat on Coles Myer.