Lightbox . . . This isn't the Private Eye Pedantry Corner - but I think we need to be clear about this. A lightbox is a transparency/negative viewer and I'm guessing that you mean either a light cube or a light tent, not a lightbox.
A light cube is the modern version of a light tent and typically includes one or more strips of LED lights. A light tent is the previous (and less bad) incarnation, with translucent walls and ceiling, with the lights placed outside of the box and aimed through the walls and ceiling. Less bad than the light cube but still useless for nearly all types of photography.
These things are basically sold to people because they're cheap and because they claim to be the perfect solution to product photography solutions. They are in fact pretty useless because of the lack of light control, and product photography is mainly about getting the lighting right.
The example photo you've shown is, IMO, terrible at every level.
If you want to photograph a watch, then your starting point is to put the camera on a solid tripod, position it to suit and then place a flash fitted with a softbox in about the same position as in your example photo (above and behind) but at an angle that doesn't create that horrible glare. The softbox will need to be at least 4 times the size of the subject but, with such a small subject, that won't be a problem for you.
But, it seems to be the watch STRAP that needs to be right for you. Same setup, same light position but it needs to be a much smaller light source and it needs to be further away. This will reveal the texture. The strap will then look good, but the watch face will look terrible, so if the strap is attached to a watch then you'll need one shot lit for the watch and another for the strap, then just comp them together in Photoshop, a very quick and simple process once you've got the hang of it.
As for using an extension tube, yes, you will almost certainly need to do that.
As for choice of lighting, it needs to be flash and although studio flash is cheaper and better for this type of photography, off-camera flashes will be good enough too. Brand doesn't matter unless you buy at the junk end.