Nice observation... "πρόσφατος" can only mean "recent". I would choose "τελευταίος", it's an adjective that works well with both "latest" and "last". I remember, back in the day when I was learning English, I had to be careful translating "τελευταίος", making a choice between "last" and "latest" based on context. Often, not an easy job! Your suggestion would also work here, albeit perhaps a little too descriptive.

I understand διαγωνισμός as being more like a competition or raffle - something you'd enter at a fete or find on the back of a cereal box. Not a competition involving any physical activity. αγώνας is more like a sporting match or some sort of other competition which requires physical activity, like a marathon, or indeed, the ολυμπιακοί αγώνες. That's my understanding, anyway, but I'm not a native speaker, just a learner - someone else might like to offer their advice.