I'm going to throw some ideas out for your consideration.
First,
it cannot be overemphasized, just how much less-stressful and productive you'll find tethering software to be, over all other methods. One static setup, you turn the camera on and never touch it again whether you're taking one shot or 300. Everything else happens at the computer. My experience is Canon only, of course, but there's software out there which will work with Nikon. I don't know if Nikon bundles it for free; I know Canon does. Getting it to work with point-and-shoots is problematic.
I strongly advise you to revise your budget upward to the level of a dSLR and dedicated Macro lens. We'll talk more about that budget in a minute. I say this because I found it rather difficult to arrange shots which satisfied me with a P&S. Not that I was unable to, but the compromises inherent were an annoyance. There was far less generality in terms of setup - I had to vary camera placement by coin size to obtain clear images of the right size and lighting was far more difficult to get right due to the very short camera-coin distance.
It bears mentioning that I am utterly and completely spoiled by the setup I *do* use, in terms of both functionality and quality. So take my opinions with a grain of salt. Knowing your rather, um,
discerning taste in coins, though, I can't help but think that a lesser solution might ultimately frustrate/disappoint you.
Now. About the money. If you end up going the dSLR route, for pricing and availability considerations I firmly believe that Canon is the way to go. The competing dSLR makers can all duplicate Canon quality in our niche, but they cannot approach the price.
There's no need to think "current state-of-the-art" for your camera. My first dSLR was the 8MP Canon 350D, and upgrading since has netted me no increase in quality. The image size increase has been utterly useless in real-world terms; I cannot display my archived 350D images full-size on a 1920x1200 monitor as it is, much less the even larger shots I'm taking now. A 350D will completely satisfy your requirements, indoors and out, now and forever. Just like computers, unless you're a professional, the lower end of the scale is still far more than any user will need, and photographing coins does not qualify as "professional-level" work.
A 350D will set you back something over $300. I can send you two
ebay links for sellers I'd buy from right now - the first is a used package, including kit lens, for $330 shipped and the second is a brand-new unused camera, body only, for $350 shipped. With a little judicious shopping, a 10MP 400D could be had in that range, as well.
The lens is far more important. Although you can make the stock 18-55 suffice, some compromises are inherent. It'll work for you if you want to approach your kit in two bites rather than one. A dedicated macro lens is where you're going to want to be, though, as soon as you learn what can and can't be done with shooting coins. Trust me on this - again, I've seen you in action.

Canon offers three dedicated macro lenses which won't cripple your bank account, in 50mm, 60mm and 100mm sizes. The tradeoff, in our niche, is only about ultimate image size and how far you can stand the lens off of the coin. All of them produce images of sufficient quality for even the pickiest numismatist. In the real world, the only difference is about how much you'll have to downsize the images before you can hope to post them anywhere online.

A Canon 50mm Macro can be had for about $265 brand-new. A nice benefit is that it will double as a decent walkaround lens, as well, something I can't say about my 100mm. The two larger lenses get a bit pricey.
OK, I've spoken my piece. You
will be able to make a P&S in your original price range create shots which will get you compliments on this forum. Increasing the investment will net you larger shots and ease of use; it's up to you to determine how important that is.
As a thought, we live close enough to each other that getting together so I can get you up to speed in person is not out of the question.