Method:
For the herb butter: Leave the butter on the counter to soften or place it in a microwave-safe bowl & microwave until malleable, 10-15 seconds. Use a fork to mash in the herbs & garlic until fully mixed. Store in the refrigerator & remove 10 minutes before adding to the filet.

For the fillets: Preheat the oven to 415°F. Remove the steak from the fridge 30 minutes prior to cooking, this is to bring the steak to room temperature & ensure your cooking times are more accurate. Season all sides liberally with salt & pepper. This can be done while your steak comes to room temp.
Add 1-2 tablespoons of oil to an oven-safe cast iron skillet & turn the heat up high, allowing the skillet to become very hot. Twirl the pan to distribute oil as it heats. Once the oil starts to smoke, place the filets face down & sear undisturbed for 2 minutes. Flip the filets & sear for an additional 2 minutes. This will give your filets a nice seared edge.

Optionally add some herb butter & a whole garlic clove, tarragon & rosemary to the pan before transferring it to the oven. The butter will melt & the herbs will season the pan drippings for a later step. Transfer your skillet directly to the oven. [WARNING] skillet may be hot, handle it with oven mitts. For rare, bake for 4 minutes. Medium rare, 5-6 minutes. Medium, 6-7 minutes. Medium well, 8-9 minutes. Remember, depending on the size of the steak, the more or less time it will take. This recipe is ideal for a 280-300gm portion.

Optionally spoon some of the seasoned pan drippings over the steak to infuse it with extra flavour.
Remove filets from the skillet & set them on a plate & let sit for 5-10 minutes before serving. This is important to bring your steak to its final serving temperature.
Top with a slice of garlic & herb butter & serve.

Notes
Cooking steaks at high temps in a skillet tends to get smokey. Open a kitchen window & turn on your kitchen’s overhead vent fan before you start to help with ventilation.

If you don’t have an oven-safe skillet or pan, sear the steaks first & then transfer them to a baking sheet lined with a wire rack. This method works really well at keeping your steaks elevated from the direct heat source but will also add additional cooking time to your oven duration. Plan on 2-3 additional minutes

Temperatures for steak
Rare: 48°C to 50°C
Medium rare: 50°C to 55°C
Medium: 60°C to 65°C

Cooking meat is a combination of physics, chemistry, & biology. Each cut has somewhat different properties based on its composition & structure, & different types of cooking will yield the best* results for various cuts. In almost all cases, cooking a steak to well done will produce objectively poorer results in terms of flavour, texture, aroma, & juiciness.

When talking about “steak”, most people are referring not to all red meat in general, but to beef from cows (that were raised on some kind of farm setting). Further, steak typically refers to specific cuts & preparations, such as sirloin, porterhouse, strip, rib eye, & so on. And in most cases, it is cooked in one of several high-heat direct-contact methods, such as broiling, grilling, searing, etc. (If it’s being baked, stewed, braised or made in any other manner, it is not usually called a steak, even if the anatomical part of the animal overlaps.) While there may be slight variation in the optimal cooking temperature for various cuts, different breeds, different cooking methods, etc, I can’t think of a single type of steak that is objectively better well done. There are many other cuts of meat (such as short rib, brisket, etc.) that benefit from long cooking times to break down connective tissue. But nothing that is normally referred to as a steak fits that description. And I’d venture to say there is nary a steak lover out there who would make fun of you because you prefer your short rib stew well done.

The techniques used for cooking beef for long periods of time are different from those used on steak, & specifically allow for the introduction of additional liquid to prevent dryness, & often employ lower temperatures so that the meat doesn’t suffer from the long cook time.

Generally, cooking a steak optimally seeks to take advantage of two major reactions, from a chemistry/physics standpoint. The first is what is referred to as the “Maillard reaction”. That’s what gives the browned exterior of the beef (& other foods) its desirable flavour. It’s usually characterized as a rich “umami” quality. If the steak is cooked beyond the optimal point, that browned portion is replaced by burnt/charred areas that not only lack the rich flavour from the Maillard reaction, but replace it with bitter, undesirable (& some say carcinogenic) straight carbon.

The second important thing that happens when you cook a steak is that the marbling fat inside the meat renders & melts into the meat. If this process doesn’t happen, due to undercooking, the meat will not be as tender or as rich tasting as it could be. This is why, to a true steak lover, ordering a steak too rare can be just as much a faux pas as ordering it well done. You want to cook it just to the point where the marbling fat renders but isn’t consumed by the heat & burned off or lost. Overcooking a highly marbled steak will make it tougher & more fibrous, as the muscle fibres tighten & the proteins denature completely. To get the most complex flavours out of a steak, you want to have the Maillard browning on the outside, the fat rendering on the inside, the nutty quality of the aging, & the mineral & sanguine aspects of the beef flesh all present. So objectively, there is a point around medium rare that is the “best” for eating & preserving all of those qualities. Eating it at any other temperature is akin to saying, “I want my steak to have less (or simpler) flavour”. If you said that about any other food, it would often bring on teasing, so why not steak?

The leaner the cut of meat, & the less marbling fat there is to render, the less the steak must be cooked to reach the ideal point as described above. So a lean cut like tenderloin is best done slightly rarer than a fattier cut. Similarly, many game animals are very lean, & extreme care must usually be taken not to overcook them. Therefore, the example of elk steaks, cited by the OP, is simply incorrect. If anything, they should be cooked less than a fatty cow steak, or they risk getting very tough, without any concomitant benefits. And when the OP mentions that “a rare steak doesn’t always imply that the steak is going to be juicy & tender”, that is absolutely true. But that is the fault of the meat itself & won’t be improved by more direct-contact, high-heat cooking. That will only make it worse, not better, as the proteins denature & the fibres contract. A tough or extremely lean cut can only be made more tender by a different cooking method that breaks down the fibres & adds moisture &/or fat over a long period.

This last point is a controversial one, & I’m guessing I may take flack for it in the comments. Most people who prefer steaks well done don’t truly do so because of flavour. There are exceptions, but in a large number of cases, the preference is driven by fear. It may fear of the health consequences of undercooked meat. It may be fear of the red juice/blood that a rare or medium rare steak produces. (Note: the red juices that are seen in a rare or medium rare steak are NOT blood. All the blood of the cow is actually removed during processing, long before it’s a steak. The red juices are mostly water, with a little bit of myoglobin from the muscle tissue that loses its redness as heat is applied, but does not actually go away.) It may be fear of the texture of softer meat. But it’s fear of some kind, more often than not. And open-minded eaters will often make fun of timid ones, just as people tease others about any kind of phobia that seems unreasonable to them. Update 6/4/16: A quick scan of the comments thus far seems to clearly support this assertion, with several commenters describing their “icky” & fearful reactions to certain normal qualities of a properly cooked steak. A few even report digestive differences, though these are highly unlikely to be actual physiological symptoms rather than cultural or psychological ones.

*When saying “best”, I am referring to maximizing qualities that most (but obviously not all) eaters would consider desirable, such as robust/complex flavour, tender but not mushy texture, appealing aroma, juiciness, etc. These things are pretty universal, & with the exception of those who have specific phobias, most people agree on what they are. Those who like well done steaks aren’t usually doing so because they prefer the meat to be tough or dry. They do it for other reasons, many of which stem from fears of various types.