My dog's an unknown brand, but I know what you mean about kids being inconsistent. I came down on my 6yo daughter like a ton of bricks when I caught her confusing the dog though, which resulted in some arguments and some tears, but she is much better now. I am really trying to involve her in the training too, so let her do simple things like telling the dog to sit for a treat, etc. She also feeds her. And then I make a point of praising my daughter abundantly when she does do it the right way. I tell you, it's more work training the child than training the dog! But I think it's all part of the package.
I talk a lot to my daughter about the strategy behind how we train and treat the dog. She now gets the whole ‘encouraging calm behaviour’ thing. The fact that my dog was a prolific jumper when we got her probably helped with that though! I still make a point of reminding her to praise the dog when she stops unwanted behaviour or just ask her to give the dog a pat when the dog is lying down calmly. It’s not easy for a child to grasp how this works, but after 3 weeks of constantly pestering her about it, my daughter seems to be slowly getting the message.
What definitely helps is to give your child some responsibility in relation to the dog but demand that they do it right. An example might be to give them the task of feeding the dog but they have to make sure the dog sits and doesn’t break their sit before they put the food down. A behavioural trainer actually encouraged my daughter to do that when she was only about 3, with my old dog. And the key was that there were no commands involved. (Because otherwise they just go “sit, sit, sit, sit,etc”) She was to hold the bowl out of reach of the dog until the dog sat down and if the dog got up as she started putting the bowl down, she had to just hold it up again until the dogs stayed in sit until the bowl was put down and she was released.
I can highly recommend that exercise. And if she alone does it, she might start feeling more responsibility and be more eager to learn? There may be other simple exercises that you can think of that she can do, especially if they involve encouraging calm behaviour in the dog.
Your dog looks like a real cutie!