I would say if you stay away from a red-toned stain you should be ok in general. If you wanted to do something grey, which would probably work well with the floor, why not a grey stain instead of paint I looked at the stain samples on Medallion, and I would probably use maple or straight grain cherry, something that did not have a lot of grain like the floor. I pulled a few which may or may not work in real life but there are close variations to some of the mid toned stains. I would just try to avoid matching it too closely or something red. On my monitor the cherry in the upper right may be too close in tone especially since it darkens. Natural cherry might be better.

The Elephant in the room needs to be talked about: the age of the kitchen cabinets/counter tops/backsplash. This kitchen is clean and functional but dated. The counter tops are tile and the floors are laminate and the cabinets/boxes are dated as well. That means the kitchen is going to be ready for a renovation sooner rather than later. I mean in the next 7 years or so. Which means you need to sit down and do some HARD thinking. A kitchen renovation OFTEN (80-90%) changes the FOOT PRINT of the kitchen (the cabs are ripped out, holes in the floor are patched, etc). That means whatever floor you put in TODAY is going to need changing or a SERIOUS repair tomorrow. Right now you are trying to get a "refresh" of the kitchen without looking down the road at the FULL RENOVATION. That can get you into a bit of trouble. In your situation, I HIGHLY recommend that you sit down and plan out the next 7 years of your renovation life. Yes. Seven years. No. I am not kidding about this. To do this type of planning you have to have an END point. You need to know the FINAL look of your kitchen so that you can work towards it one step at a time. That means you need to have picked the COLOUR PALETTE for your kitchen before you lift a hammer. I like to see homeowners shop around for samples of things they like and then put those things into a basket in the trunk of their car. They will then keep the basket handy for when they see a 'GOOD DEAL' somewhere. If the deal fits the colour palette PERFECTLY, they you purchase it. If it doesn't, then you move on and save your money. Whew. But to answer your question, your wish for gray/white floors with red oak is going to be EXTREMELY tricky. Gray floors can have a peach/green tinge to them. And that will be a MASSIVE clash with orange-brown cabinets. You are welcome to put the floors in the house and kitchen, but BE PREPARED to HATE what you see. That's OK. Hate is an excellent motivator. It will move your 7 year renovation time line forward to include a 3-5 year time line. But your choice will push you towards a new look very, very, very quickly. Before you do that....please make sure you have a path set out to get to your final destination. Once you pick new floors, the destination will come faster than you ever thought possible. You will notice your inspiration kitchen is white on white on white. It is airy and sophisticated transitional/modern design. Your kitchen is not. As cute as it is...it is not. Please take a moment to investigate WHY you like the photo so much. It may not be "just the floors". You may LOVE the look that has been created. And that means a total redo of your kitchen.

What's the question?

You need to put both samples on your floor for review. Looking at photos on the internet or getting opinions from anyone else is far less effective than seeing the actual samples on your floor.