<<While I was on base leg the other aircraft announced that he was also on base leg from a "straight in join on base",>>
This non-compliance with Rule 12.
<<and as he was lower than me he was landing first>>
This is no excuse at all and holds no water.
IMC1, your conflicting aircraft has not conformed to the pattern of the traffic applicable at the time. The direct base join is no different to a straight-in approach into an uncontrolled circuit. It is not an intelligent join if the circuit contains conflicting traffic. I think that you were right to feel aggrieved.