In my view (preference) the lights should be in the ceiling where they can illuminate the counter tops and be outside the user's direct field of view unless he looks up. Shadows from objects and users should be kept in mind. However, this seems to best apply to more conventional 8 - 10 foot ceilings. I suppose pointable light housings mounted to exposed beams might work in a cathedral configuration.

In addition, small lights under cabinets will help.

I'm surprised that your electricians would bid on a job without an electrical plan.

Not sure about lights in a vaulted ceiling but your fridge is in a very bad spot - too far from sink and island in the way. do you have a whole floor plan? If not much else can be done, at least move the fridge to the other end of that cabinets run.

They make new LED lights for vaulted/angled ceilings. The lights scan be adjusted to direct light where you want it more. Don't let them install plain old recessed lights designed for horizontal ceilings.