Interesting and relevant, the boys do Ju-Jitsu (I don't know of the exact type and it doesn't really matter to me - but they recently went from being independent to part of the Gracie Jiu Jitsu group) and our girl does gymnastics, all at local clubs.
The boys at Jiu Jitsu:
Have a really good instructor, ex-copper, has young kids of his own, is firm but fair and engaging, has a sense of humour and all the kids respect him. Has some views on vaccines that I don't agree with but I can look past.
Started as pure fun and gradually introduced concepts like respect & discipline (which I like)
Has very much increased their physical strength and ability to stand up for themselves if required.
I've always 'rough-housed' with the boys and play wrestled on the floor - their technique and ability to resist being grappled is really developing. If the 10 year old puts me in a hold, there are times when it's genuinely hard to get out of, and I can't easily chuck him around (playfully) like I used to.
The boys also play football. I think the Ju Jitsu helps a little with the strength, resilience and coordination.
Our girl at gym:
She is SUPER into it, and it's easy to practice at home (she's always cartwheeling and handstanding)
Staggeringly strong core and solid tummy muscles for an 8 year old. Very flexible too, they are practicing doing the splits.
She has the best balance, and does better than the boys at things which require balance and coordination like surfing and skating. She's not as strong or physical as the boys but has far greater fitness (stamina and endurance) and coordinated movements.... TBF our boys are a bit lazy though.
There aren't many boys in this gymnastics class, which I think is a shame. Our boys did 'kindy-gym' for a while but were never that into it.
She was never that into the rough-housing type of play but we do fun things like practicing handstands, lifts, seeing who can hold a plank position for longer (she can), or even standing on my shoulders and pretending it's like a circus thing.
All the kids do swimming too, and the girl (8) and boy (10) are now at the same level in the same class, much to the older boys chagrin. Girl has better technique and ability even though boy is much stronger physically. I think gymnastics and swimming complement each other better than jiu-jitsu and swimming.
All in all any activity is better than none, and there's a lot of choice. Be mindful of the cost to yourselves, not just financial but also the running around, dropping off/staying to watch, going to competitions etc etc. It becomes a lot, and I find it's very easy to overcommit.
at D's age I think the quality of the instructor and the class is going to be waaaaaay more important than the difference between karate and taekwando. I'd honestly go with "instructor you get the best vibe from"
This is really good advice, a good instructor makes a huge impact and different kids will respond to different types.
I'd add to that the size/make up of the class, and even how comfortable the venue seems, the times of the classes, how easy it is to get to - all of that is part of the vibe.