As a kid growing up, Mom had 2 mini Poodles, Fancy & Petty. Fancy was feisty, very good little watch dog, keenly aware of what belonged on the farm & what didn't. Petty was more laid back. I could play dress up with her. She wore my baby doll dresses, bonnets on her head, lay in the doll stroller, etc... Fancy on the other hand liked to hunt mice. She was busier. While I would sit & brush her out, she wasn't as patient with grooming. Our Doberman would get her down & pull stickers out of her curls & she would sit for that for hours. I guess I was too gentle. Silly Poodle. But she was also the dog who learned how to drive the neighbor's cattle or the hogs out of Mom's garden ( a very dangerous job for a dog so small). She learned how to be sly, sneak into the midst of them, bark like a mad thing & she'd race around, snarling & snapping & scare the animals off & of course the Dobermans would back her up. Both dogs, highly intelligent (Fancy more so than Petty but Petty was no dummy), both of the same breed & similar breeding, both raised & handled by the same people. Just totally different personalities.
My Standard male is almost 2 years old now. He can be very calm & laid back in the house but with the snap of the fingers he can be ready to romp & play, go run outside, annoy the squirrels & terrorize the ladies. He can be a prankster. At this age there are times when he'll crate himself & I can hear the feet tippy tapping & this is his sign to me that he's got energy & he needs to go out & blow it off. He'll go terrorize the squirrels or I'll take him out to run with the other 2 working dogs we have. He is also gentle enough to live with 2 very small Chihuahuas. There were pups in his litter even more laid back than he is. One beautiful pup that I believe was silver & white (or would be silver & white). He was so chill I don't know what it would take to get him excited. In the same litter my boy had a brother who was more alpha, a little hotter, needed a certain type of home. Nice pup just temperament very different than most of the litter. Good breeders know their pups & will be able to advise you.
A lot will depend upon your children & the personality of the dog you choose. You know your kids. If they're rough & tumbles & full of energy & would like to play chase games & play ball, etc... then you'll want a calm dog who is playful & who will enjoy those types of games & kids. Of course then you work with the incoming dog to learn how to safely play with the kids. On the flip side if you have the quieter kind of kid or particularly gentle children you'll want calm & more mellow. There is a bit of an art to matching up the right dog for the energy of the whole family so you'll want a breeder who listens to & understands your needs no matter which Poodle you choose.
You will probably be able to find most of what you want in either the mini or the standard. There are even breeders breeding smallish standard Poodles. My bruiser is probably around 75 pounds & he's getting more mass on him these days. That's one thing you can't undo is the size.