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"Buy Market" = buy at the best available price. No guarantees on any specific price. Get me in/out.
"Buy at the ask" = a buy limit is placed at the current ask price. Guarantee is that you will get that price or better. However, if you click too late, the market can run away from you if you are too slow and you will get a bad price.
In super fast or thin markets, often better to place your limit to buy at the actual price you want, and if it's equal to or higher than the lowest ask you will get a fill (at least partial).
You are exactly correct. The end result would be the same, assuming a static market; the difference in practice of course is that as you press the "buy 3 @ market" or "buy 3 @ ask" buttons, the market can move higher because offers are pulled or taken by someone else. That subsecond delay between when your brain gives your hand the signal to press the button and where the market is when your order arrives may cause a large difference in what you think you are buying and what you get. A guaranteed way to get your price or better is to "buy 3 @ 144.57" -- of course then you might not get a fill if the above scenario occurs and you are left with a resting order to buy.
To be clear, your examples are right on, in the most basic scenario of market mechanics. When you introduce hidden quantity orders, microsecond HFT, and all the other realities of the market, other scenarios are quite likely to occur. For example, your buy 10 at the market is completely filled at 57, meaning there was a refresh offer there, and that he probably has more to sell--not the best scenario to be in.
