Do I understand that right?
Yes. Match
changes the handles for all selected keys shown in the Curves view when any handle is moved. The Curves view doesn't change key values at all.
For the Separate
button, hover or press F1 for the tooltip:
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Changing curves will "separate" the handles, so the tangent is broken. This creates a cusp. It makes Spine behave similar to how it did before version 4.0. Enough people needed that, so we added the button, but I don't suggest using it in most cases because it's usually better to avoid cusps.
The tangent is the angle of the handles through the key. Here the curve handles are not separated:
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The value of the curve is smooth through the key. Here the curve handles are separated:
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The value has a cusp at the key, meaning the value changes abruptly. It can be seen more easily when zoomed in:
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Careful not to confuse separating curve handles with separating properties that are normally keyed together (X + Y, scaleX + scaleY, shearX + shearY).
You should view your curves in the Graph to more easily understand how values change over time, especially when starting out. The Curves view is more advanced and is most often used for editing multiple curves at once. Spine used to only have a Curves view. While it's a powerful and unique feature, it gives a very narrow view of the animation. It's easier to understand the curves the Graph view and easier to see how the values change over the whole animation. Also the Graph view can change both key values and curves.