I just revisited the image data and it seems that the OP is using an aftermarket (read not Nikon) lens. Or at least a non 'computerized' lens, AIS? Maybe there is an incompatibility issue? Although I don't see how that could change the metering mode, we are talking "computer" here.

Since this is the 2nd camera the OP has had with the same problem, for it to actually be the camera making changes without operator input is VERY unlikely.

I'll have to give that a try :)

If you live in a gray in the winter area, photograph a white image (word page) your computer monitor at a bright computer setting. Don't worry about focus or camera movement, for this it's better to be out of focus and lens movement too.

PS This image was taken with a D600 using the kit 24-85mm lens

(Download)

I've been using a D600 (basically the same camera as the D610) for about 3 years and it is a gem of a tool. The sensor is the same as what is in the D750 and except for a newer processor in the D750 (which accounts for the 750 having better low light ability) the D600 series will produce comparable images. For a considerable saving your grandson can be shooting with a quality camera and will have enough left over to by a better grade of lens.

I can't believe that 2 D750's automatically changed metering modes, I strongly can't believe that if there were 2 D750 exhibiting the same strange behavior that both of them went to the same customer. The odds of that happening have to be astronomical. And even though I do not have a D750 I can't believe that any camera, let alone the 750 would do this without operator input. I think the user here is under the impression that he should be getting the same results (image wise)no matter which metering mode he is using and has been attempting to 'fine tune' the different modes to give him that result and since he can't get there,that is what his complaint is.

If you "download" and save the images you should be able to access the data either by checking the image properties or by loading them into Photoshop.

I think this story belongs in this thread (and others like it)
A family member has an expensive camera($6K+ when new) that had a dirty sensor when he last used it. He put it away while he was gone on an extended trip. He lives in a very humid area and when he next when to use the camera the dust had adsorbed moisture and that turned to mold. A very expensive repair. I know we tend to procrastinate with sensor cleaning and use Photoshop or equal to remove spots but, please don't wait too long it can bite you HARD!

How about you take a series photos with out changing anything between shots and see if the images remain constant.

I for the life of me can't under stand what the problem is.
When the metering method is changed the image is going to change. When you adjust the fine tuning the image is gong to change. The camera is responding to the operators inputs and the way this thread reads, he don't think it should be?
And what in the world does fine tuning the focus have to do with exposure?
It seems like a matter of a confused user. Or possibly the user can't articulate exactly what he is doing?
Which ever I'm sure the camera is fine.

I didn't get into rallies but loved time trials. Thought I was going to be the next Stirling Moss or Roger Penske. HA!
Little talent + lots of desire= broken car and wallet

With out additional information, it's impossible to tell. If you're not changing anything between shots there will be very limited difference from one shot to the next as long as all things remain equal. With the D750 images you were making changes between shots.

1962 MGA 1600 mkII. During the halcyon days of my youth.

I agree with what Rick36203 said. I downloaded and looked at the data for the 3 pictures. You used 3 different methods of metering with no other compensation. When you do that with out making any adjustments you will get different results and what those 3 images show are about what I would expect the results to be. I think your camera is perfectly OK.
You might want to read up on what conditions the different metering methods are designed to be used under.