I think rebound and bounce are the same, though I'm not sure everyone uses them the same way. Here's what I see:
There can be "real" causes and "perceived" causes (liver training). From what I have read, rebound can be acute (one time), or chronic (repeated acutes that leave their body in a constant fighting stance, so you get high flat numbers ongoing).
"Real" rebound is from actual low numbers or steep drops.
Liver training rebound is when they get safe numbers that they are not used to, and their body reacts to it as if the numbers are unsafe.
Liver training would never get to a chronic pattern though, as it goes away on its own as they spend more time in good numbers. So liver training is only acutes, real rebound can be either acute or chronic.
I really wish we had 2 separate terms for them, it is just such a confusing mess whenever it comes up. Maybe some are trying to do that with bounce vs. rebound? I'm not sure. I read them as the same - a rebound/bounce response. Then it's the puzzle to figure out is this real rebound (meaning the dose is too high), or liver training rebound (where you want to hold the dose).
I think those nadirs are great and you wouldn't want a steeper drop, so I would not increase the BID dose. If you wanted steadier lower numbers, I would consider shooting earlier, either TID, or something like shooting at +10s, or as needed shooting (picking a threshold like 150 and going ahead with another shot as long as they are past nadir and rising again). Any of those might require dose reductions, you would not necessarily shoot the full BID dose on an early shot (you probably already know that, just mentioning it for clarity for any newbies reading along).
When they stay in greens to +12 or longer, it can be too much insulin, or it can also be a sputtering pancreas. Hard to tell which. To some extent you can guess if they shoot up really fast at a point like +14 and go from greens way up to reds or so in an hour, that is the insulin wearing off. If they gradually drift up more slowly (like Bix sometimes used to go 24 hours or longer before he hit something like 150 or 180 to give a shot) and don't zoom suddenly, that's possible pancreas sputtering. Not hard & fast rules, just some thoughts.
I think it would be fine to stay where you are for a bit since you are seeing good nadirs, or I think it would be fine to try some version of early shooting if you are feeling impatient. Nothing wrong with doing that as long as you do it safely by regular testing for early cycle numbers and nadirs, hypo toolkit on hand, and adjust the dose as needed based on what kind of overlap you may see with early shots. It's up to you really what strategy you want to take, how your schedule is, etc. etc.
When I got impatient and tried early shooting I really didn't get anywhere and I think it slowed us down (Bix got good duration so it turned out he didn't really need it, plus I tended to overreduce the dose). but I've seen others make good progress that way. Sometimes even doing early shots as going for TR over a weekend or something can help promote pancreas healing and then you can go back to BID during the week and get better results. If you can get the dose right and get good numbers that can be a good strategy.
That is great he is symptom free! YAY!