Luring as San demonstrates in the video is a very effective technique. You can do it from a stand or a sit, and I'd recommend both, or you'll have a dog that only understands it one way or the other. Keep the treat in your hand until she's dropped into a down, then immediately mark it and give her the treat. Release, repeat.
If you do use the lure method, make sure that you only have a treat in your hand as long as you actually need it. Once he's consistently following that hand into a down you should switch to doing the same motion but with an empty hand, and then treat from the other hand. I can usually get to this step in the first or second short training session. If he looks confused at first when you don't have a treat in your hand anymore, give him a few seconds to figure it out. From there I gradually fade the motion so it's much smaller, from a hand moving down to the floor to as little as just a finger pointing down. That is now my hand signal for the down.
Until the dog is consistently responding to my hand signal (lure motion with an empty hand), I'm not even worrying about a verbal command - there's no point in saying "down" until he's actually doing it. Once the behavior is consistent, you would say the command before
you use the hand signal, not at the same time. If you say it first, it will become a predictor for the hand signal that follows, and become associated with the behavior. If you say it at the same time, he'll be paying attention to what he knows (the hand signal), and not what you're saying. Gradually I start waiting longer and longer before using the hand signal after giving the verbal command, letting the dog think about it and figure it out on their own, until I'm finally only using it as a reminder, if necessary, to let her know what I want.
I prefer not to physically manipulate the puppy into position, because as you have noticed, they may think it's an invitation to play. Another very effective method that I use in addition to luring is to "capture" the behavior as the puppy offers it, then marking and rewarding. Remember that you're not actually teaching her to lay down, she does that all by herself, probably dozens of times a day! But if you mark and reward her when she lays down, she's going to start laying down more often. You can mark it and toss the treat so she has to get up and get it. Wait for her to lay down again, mark and reward. Repeat.
Once she's consistently offering a down, you can say the word right before she does it - mark/reward. That will also help associate the word with the with behavior.