How do I create a more natural look for watch photography using a Light Box?

You've done a great job for being 'fairly new'. I agree with Steven in the things he said, so experiment with those.

Edit: Your focus, depth-of-field choice, composition and exposure are spot-on.

Here's what I can see that really stops the shot from being where you want it:

• There's a green cast. This is probably from the type of lights that come with the kits. If your camera has the ability to white-balance, use a white card and color balance before shooting instead of relying on auto-balance. This also causes your blacks to shift, which adds to the next point.
• The contrast is too thin for a product close-up.
• There is lint/dirt/fibers/spots in the background. Although they are individually very small, the brain picks up all these things and it feels 'like something's off'.

Also be aware that the blue in the example shots provides natural contrast to the brushed silver watch, which is helping that photo 'pop'.

Here's your shot after some correction in photoshop:

What was done here was color-balance correction, black-level correction, some 'de-hazing', and a slight increase to the highlights. All these adjustments were fairly minimal, except the black-bump, which was a good bit.

The background was cleaned up with a spot-healing brush. Because you mentioned that you wanted to preserve some imperfections, I only healed/cleaned up the background and left the watch untouched.

As far as the reflections in the watch face, its tough. Use a polarizing filter to cut the surface reflections down. You will want to play around with it a little if you want to preserve some of the light reflections on the face. Also, try moving the lights at different angles to each other, and if possible, different power levels.

A light tent is probably the wrong tool, here, but it's what you've got. You're definitely right that it's the lighting. A light tent is made to diffuse and soften the light but the flip side is that soft light doesn't reveal textures very well. With product photography, it's all a game of angles in terms of getting a surface to reveal its texture in a photo. And a professional is more likely to use lights with modifiers rather than a light tent, so they have more control over those angles, placement, and diffusion.

But hopefully you can adjust where the lights are outside the box. The example images you post, the light is coming from an angle from probably up above camera left (look at the shadows) not from the front or top of the subject.

A hard light from the side is actually going to really pop a texture, because shadows define shape to our eyes. And a specular highlight can indicate something is hard and shiny, but can obscure stuff.

Also, angle of incidence = angle of reflection. If you were able to shoot from another angle, you could be outside of the reflection on the watch face, or if you rotated the watch slightly away from you, you wouldn't be in a mirror situation. You could also consider trying a circular polarizer to see if you can eliminate/reduce the reflection, but then you might lose some texture on the rest of the watch.

Product lighting is not subject that's easy to cover in a short stackexchange Q&A. There's an entire college textbook on this subject, Light—Science & Magic: An Introduction to Photographic Lighting of which the 6th edition is coming out April 28, 2021. That book pretty much covers everything you need to know about product lighting and finessing angles for direct/specular (shiny) or diffused reflection (not shiny).

Several questions on your questions...

"A more natural look"

First of all. Photography is not just about light, it is also about shadows.

On a reflective object, it is not just about reflections of light sources, but also of dark surroundings.

As we do not live in a "heaven-like place" with light, and just light all around us, a lightbox looks not that natural.

So, my first answer is, add some "negative lights". These are black cardboard or paper in some places.

Try adding one full wall or two and see where this black is now shown on your watch.

"The photo is too dark"

Do not use auto exposure. Use manual mode.

Having the right exposure to a completely white place is about knowing the dynamic range on your camera. You have to know your histogram. Let me explain.

If you have access to a gray card (you could DIY*) you should have on your histogram a strong line at the center of it.

Using the same exact exposition now take a picture of a white sheet of paper. Stack two or 3 of them so you have a "whiter" target.

Now, look at your histogram. That is where that white on that light condition should look like.

The histogram of a reflective surface is different, because you can actually blow the whites on a reflection, and that is totally ok.

Do the same with a really black cloth, like black velvet. Now you will have a spike on the black on that light situation.

Now you have something to compare the histogram of the background of your light tent using that exposure.

Of course, you can just add one or two stops to your shutter speed.

• A DIY gray card can be done by printing a checkboard pattern. Try let's say squares of 2-3mm

Of course, it will depend on the quality of the printer. Try using laser-based print with a really good deep black toner using extra white paper.

In time use this technique to find the right exposure using white paper or a white wall and then compensating the f-stops so your photos are not dark.

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If you want to fix the dark look in post, you can blow some whites on your histogram. Just keep the white on the background natural.

Te reflection of yourself on the watch cover

Take a look: How to shoot a reflection in a ball bearing without appearing in it?

In your case, remove the white wall on the side where the camera is. Now turn off all lights outside. Move the camera further away and now you will reduce that reflection because it would be less stuff to reflect on in the first place.

Not asked

Watch your framing. The first image needs more room on the top.