Dark floors and not a lot of natural light in that space. I’d go white personally.

I’d go white. I’d love to see more pictures of your home.

I was ready to say white cabinets, but I really like that sage green cabinets above. Sage green (with red and other bright splashes of color) is so cottage-y, and I think the soft pop of color would be superior to white. I'd keep the cabinets very simple. Your knotty wood and the angle are already major players, and if you bring in kitchen items that're "too much", you could easily reach the too-many-clowns-in-the-kitchen stage. I'd aim for calm and simple in all things kitchen.

Absolutely no to dark cabinets. That'd be the worst possible choice.

I like the butcher board countertops in the above picture, but you already have so much wood in the space. For the reasons mentioned in the first paragraph, I'd go with as plain a white countertop as can be. White cabinets will give you a tie-in to the white trim. I know that tile is O-U-T, OUT these days, but it's affordable and screams cottage. If you go with a tan grout, it isn't awful to keep clean.

Are you doing a backsplash, or will the wood be your backsplash? If you're doing a backsplash, I'd vote again for something simple and white. Penny rounds are the epitome of cottage, and they're quite affordable.
What concerns me in the picture is the loft space above the kitchen ... is that glass (which would read modern-modern-modern), or is it a space where railing isn't yet installed? Simple white wooden railing would look so nice up there ... and would tie into your doors and trim. Of course it also needs a pretty quilt (with plenty of red accents) hanging over the rail.

Because your space is open, the color is going to have to work with the rest of your decor. Without that info, it is hard to give advice.

A friend just painted her cottage kitchen a really pretty blue, slightly lighter than navy. She's going to add marble counters and has stainless hardware. It looks very fresh and cheerful.

Not the exact kitchen but very similar to this photo.

I dont like white with the natural wood on the walls. I think the sage green that someone posted would look really nice or maybe a gray blue.

how tight is your home.cottage can have variable humidity and temp ranges with your absence...water nearby...etc. I'd never do painted. We

have old knotty pine.....a friend in Wisconsin on a lake has a style very close to yours with lots of wood trim/etc and used natural hickory 5 piece fronts. It looks great.....there is a lot of light cool tone in natural hickory....she put glass door uppers to show things

I do like the sage green suggestion. Not a big fan of more wood on more wood on more wood.

I too love the sage green suggestion with a white countertop.

This is a case where sample doors should be brought in. I find your wood very pale and neutral as is shown. The sage green to me would be lost in the room. If paint-go more intense/navy? Barn red? with some white mixed in. But putting up a few different doors will reveal these kinds of things.

And definitely find the town in your area where design showrooms exist. Some extra special creativity lurks in vacation/resort areas and they do kitchens beyond imagination. I've seen amazing as well as simple yet stunning things in these places. Susan Serra was in northern mich... now on Long Island but big internet presence too. Just an example.

One more vote for sage green, then white if you must. I love your cottage. It is so charming.

Given the choice between only white and dark, I'd go with white.

If the choice is between light and dark, I would choose aqua.

Something like this:

http://interiordecoratingcolors.com/wp-content/uploads/2018/02/kitchen-cabinet-paint-colors-pictures-ideas-from-interiordecoratingcolors-intended-for-kitchen-cabinet-paint-colors-explore-possible-kitchen-cabinet-paint-colors.jpeg