When I think of "sport" I think of "fitness" watches from Garmin and Apple Watch. Patek sport watch is definitely not that. What is meant by "sport" in case of the Patek sport watch?
A dress watch is thin, almost always with polished rather than brushed surfaces, rarely with a date complication, rarely with lume, sometimes without a second hand, with a leather strap. It’s meant to slide underneath a shirt cuff and be both elegant & discrete. It is probably not water-resistant. It may not have a minute track.
A dive watch is meant for scuba diving. It will have one-way bezel marked with minutes for tracking dive time. It will be heavily lumed. It will typically have a metal bracelet.
A field watch is based off of a WWI officer’s watch. It should be reasonably water-resistant (weirdly, many are not!). It will always have three hands, and must be hacking (means that the second hand will stop when you set the time, so that it can be set to the exact second). It probably has 13–24 in an inner circle. It definitely has a minute track.
A Garmin or Apple Watch is … not really a watch, but is a wearable computer.
Yeah, the line about them is that they are ‘desk divers’!
I have the idea that plenty of folks do still use them for diving, but of course I could be wrong — it’s not a community I am much around.
They are more casual watches. They will have a seconds hand and glow in the dark hands and markers. They will be able to tolerate getting wet (ie no leather).
They started that way, but not so much now. Still, that’s the lineage they fall under in certain contexts.
Sport watches started with things like the JLC Reverso, which was designed to meet a challenge of a watch that could survive a polo match. The case includes a swivel that lets the dial flip around to keep it protected while playing.
Quartz watches upended watch makers across every price point, totally destroying the bottom of the market. The survivors moved towards mechanical watches as luxury items, and even the holy trinity of Audemars Piguet, Patek Philippe, and Vacheron Constantin had to respond. Audemars Piguet revealed the luxury sport watch Royal Oak in 1972 and there wasn't any going back. PP designed Nautilus in 1976 and Vacheron brought out the Overseas in 77. These 3 lines make up a huge chunk of the holy trintiy business now.