Update sync split-button example with changes in Explainer doc

[dandclark]

Update select implementation so that the controller code is applied to the button and listbox parts directly, rather than using the <slot>s.

Refactor so that the same code adds controller functionality to both the default parts and to the user-replaced parts, which should make it easier to keep these in-sync.

In the split-button example:

• Remove the preventDefault that was keeping mouse clicks from working on the left half of the button, and achieve this functionality instead by moving `part='button' to the arrow element.
• Add slot/part to listbox.
• Fix typo in CSS class.

[gregwhitworth]

I love the changes to the controller code. You'll need to put the part='button' back onto the button-container or come up with another solution for it to match the bootstrap example (I linked to it in the code).