I do not see how your mechanism is beneficial for a couple of reasons.
First, when you place a limit order, it is an upper (or lower) bound. If the market maker can improve upon it, they are obligated to do so. It is also in their interest to do so because you are allowing them to substitute your liquidity for theirs. You are adding liquidity to the market and it is in their interest to reward you for doing that.
I think part of the problem is that you do not seem to have an actual reservation price. If you had a reservation price of 22, then an ask of 21.90 puts you ahead. If your true reservation price is 21.80, then you are worse off by raising the price to 21.81.
It sounds like you are wanting it to fill, but you want to feel like you won something extra. I think you are overestimating your role in the transaction.
As already mentioned, if the price continues to climb, you will be chasing the price increase and exceed the 21.90 that you could have had. However, if it falls, it isn't likely due at all to your actions. Indeed, you may be working against your own interest.
If the price falls, it is because someone else is selling shares, maybe a block of shares.
If the dealer wants to buy inventory, they will use your price as a floor. When you raise the price, they will raise the floor. If they get a large order, then they will invade the floor and take your order. If the dealer needs to unload inventory, they may take your price at the first opportunity if there are no intervening orders. You may get your 21.80 if you are simply patient. It may also never reach that point.
This discussion gets a bit more complex if you can see the order book because there are a number of shenanigans played in order books by other participants such as iceberg orders and so on. Indeed, you seem to be trying to play shenanigans but I don't believe you are having any impact.
I think you are seeing this as if you are approaching a cliff edge, trying to touch the edge without falling over. The part you may be missing is that the cliff edge is moving. At 21.80, if you could see the future, you may find that you are way over the cliff edge already and that by altering your order to 21.81 you lost a penny. The difficulty is that you cannot see the counterfactual case where you did not alter your order. Your system may well be losing you money, but as an opportunity cost so that you cannot actually see it happen.
I don't think any of us can recommend a strategy to you because we do not understand or know your goals. Your goals should drive your strategy. The rules that you use to accumulate a large position will differ from those attempting to engage in day trading. The other problem is that this forum does not support mathematical notation, so potential strategies to conserve cash flows cannot really be discussed here. You could try reading on the Quant SE.
This strategy doesn't have a name because its function appears to be emotional and not financial. I don't think you could systematically improve on the bid-ask spread with this method, even if you were a mega trader.