If you "saw" on your horse's mouth by alternating squeezing and releasing with your hands, you're riding your horse from front to back. He might look like he's "on the bit" because his head is down and his nose is on the vertical, but you don't have an honest connection from back to front.
The only part of your horse's body that you can affect by "sawing" or vibrating both reins is his jaw. Moving the bit in his mouth encourages him to chew. When he chews, he flexes in the jaw.
Vibrate or squeeze on the inside rein to keep his neck straight, and encourage him to chew and thus flex his jaw.
When you just flex the jaw, you might think your horse is on the bit. But then you wonder why he comes off the bit when you ask him to do something like a transition.
The reality is that he was never on the bit to begin with. All you had was a flexed jaw.
To put your horse honestly on the bit, use your "connecting aids". Close both legs to add power from behind as if you're doing a lengthening. When your horse "arrives" at your outside hand, close that hand in a fist to capture, contain, and recycle the power back to the hind legs. Do this for 3 full seconds.
THEN, lastly you can vibrate or squeeze on the inside rein for two reasons:
To keep his neck straight. Your goal is to keep him from bending his neck to the outside in response to your closed outside hand. This means that when you ride with his soft (hollow) side on the inside, chances are you won't need any inside rein at all because he won't try to look to the outside when you close your outside hand.
To move the bit and encourage him to chew so he flexes in the jaw. You only need one hand to do move the bit. So never do with two hands what you can do with one hand (move the bit). That way you have the other hand left over for the more important job of recycling power back to the hind legs.