The term 'Derby', refers to any style of footwear with a 'open' lacing system, where the facings are stitched on top of the vamp meaning they can be opened outwards, as such, are suitable for a wide foot or a foot with a high in-step.
The word 'brogue' refers to the punched detailing, which at one point was used to let water out of country shoes; these days, they are purely decorative. Semi brogues are an excellent halfway-house between formal toecaps and decorative brogues. They can be dressed up or down offering a similar level of versatility to other brogues.