I start with a bit of background information, and try to give a "recipe like" answer at the end.
In "technical" terms what is missing in your training is what is know as stimulus control. In the case of a sit:
- the dog sits only when cued to sit;
- the dog does not sit in response to a down cue;
- the dog does not sit when not cued (*);
- the dog does not do something else in response to your sit cue.
(*) this applies only when the dog identified that he is in a "working" mode; he's of course allowed to sit in any other situation.
So you'll be done with your training only when you have perfect 100% stimulus control (this will never happen). So you have to practice until you reach a level you're happy with.
Visual and verbal cues
When training a new behaviour you first have to teach the dog how to perform the behaviour, then you progressively add stimulus control for your visual cue and then for your verbal cue.
Dogs are experts at reading body language, so then will learn the visual cue more easily, and when teaching the verbal cue you can use the visual cue to initiate the behaviour.
It is important to teach the visual cue first (without saying anything then), and, only when the behaviour is under stimulus control with the visual cue, you can add the verbal cue.
So it is important, when teaching the verbal cue, to remove any kind of body language, so that the dog focusses only on the verbal cue. Give your verbal cue "sit", without moving, count to 3, then give your visual cue, the dog sits, you reward. Progress slowly, at some point you can remove your visual cue, the dog will respond to the verbal cue only. If he doesn't, give go back a step and give your visual cue.
Many people (including me) would be surprised to see that their dog actually responds to visual cues (body language) they are not aware of. Apparently my dog can read my lips... Of course he can't but I realised yesterday that when I say couché (usual cue for down in French) I have a facial expression that he reads easily.
Positive reinforcement training
Positve reinforcement training is a training method based on a scientific approach of how dogs learn. Long story made short, it doesn't use punishments, we focus on behaviours only (and not on the dog supposed dominant attitude for example) and it simply reinforcement behaviours we want and ignore those we don't want.
In practice you proceed in three steps:
- You get the behaviour (different methods exists, for a sit and a down, luring with and then without food works well)
- you mark the correct behaviour, either with a clicker or with a word (short "yes")
- you give the reward
How should you proceed ?
If your dog knows already how to sit and lye down, I would proceed like this to "clean" your set of signals:
- Assign a clear verbal cue to the "sit" and the "down" (usually sit and down, but make sure everyone in your family uses the same).
- Choose a visual cue: make it clear and don't mix it with excessive body language, eg. one finger up for a sit, but don't lean over the dog. When you train the dog, ask someone to check that your visual cues are not mixes with other less obvious (to humans) signals that the dog could interpret
Then you just start over your training, based on positive reinforcement. You cue the behaviour, if it is correct you "mark" the correct action (clicker or verbal signal like a short "yes"), then you reward the dog. For sit and down a food reward is perfect, it allows you to proceed in quick, fast paced, 5-minutes-long training sessions, once or twice a day.
So when you're in a relaxed mood, go in a room, alone with the dog. Then proceed first with sit. Second session with down, next ones you can mix both.
Assuming he already has basic knowledge of the cues: give your cue, as soon as the dog initiates a sit, click (or say "yes") and reward. Progressively "click" and reward only when he sit completely and quickly. See above for the distinction between the visual and the verbal cues.
Once "sit" and "down" are OK with these criteria, you can start mixing both, and also adding other "neutral" stimuli, eg. saying "banana" (if the dog sits or lies down, don't reward and move on), wearing white socks (but please don't do that), with another person in the room, etc.
As dog don't generalise well, you'll have to train in different situations, rooms, adding distractions, etc.