I guess I'd say I'm casual, although the way I play doesn't feel casual. But I don't compete or battle with others.
I play through the story, but my main goal is catching and evolving all of the Pokemon. I won't leave an area until I've caught every available Pokemon there and I make multiple teams that I switch between to evolve them all myself.
It's a very obsessive and lengthy process when I play a Pokemon game.
I'm somewhat casual, I mainly just look for shiny Pokemon. But if I'm doing Masuda Method breeding for a shiny, I tend to also go for natures and IVs... this can be seen with my current Shinx hunt in Alpha Sapphire, where I have egg moves, Jolly nature, IV & Shiny Breeding.
I do some competitive from time to time, but it usually ends with me being beaten horribly. I have found some rather fun sets I used...
Breloom with Poison Heal, holding a Toxic Orb, Adamant Nature, max attack, with the move Facade was one of my favorite sets in Alpha Sapphire.
I also enjoyed a Volt Absorb Lanturn in Ultra Moon...
EDIT:
I also do sort of challenge runs, never super competitive, just different ways of playing. My current Platinum playthrough is a mono-electric team. My Luxray doesn't have the best nature, but I love her too much. Yes, being a mono-electric team means the ground-type E4 member will be a challenge- and let's not forget Cynthia's Garchomp, and the fact that Platinum has scaled up difficulty compared to Diamond and Pearl- but so far it's been fun.
The main above still stands: I like shiny hunting. In fact, I've done several Let's Go Eevee playthroughs with only shiny Pokemon, which has been fun. Hunting can drag on with bad luck, but for the most part it's still a blast.
Casual. I prefer to build my collection and do basic Type Matching battles (I.E. Water beats Fire). I wish there was a second battle facility in main series games that focus on that casual level of battling to be honest. Sometimes I feel like my Pokemon just sit there after I finished the game :'(
I am casual. When I play, I finish the game and then go on to complete the dex. If there is a battle facility that gives proof that you beat it, be it medals, stickers, statues etc, I try to complete that as well however with varying degrees of success.
As for competitive, I used to do a lot of battling with my brothers or friends back in the day and it was fun. But since everything has moved to online, I have no interest in battling Johnny Anonymous from the other side of the globe because the human interaction just isn't there.