Personally for me think they look beautiful either way. BUT, and it's a big but, I wouldn't want to have to worry about keeping a rug under a table clean. I may love to vacuum, but I it's just not so fun when dodging table can chairs under a dining table. It's so much easier to sweep a bare floor in this situation. But that's just me.

The shape of an area rug is certainly a matter of preference or personal decorating style. One of the most important factors in selecting a rug for a dining room or any 'eating' area is the size. Where possible, try to make sure the chair legs do not get caught on the rug when sliding in and out. Another important factor, who will be dining with you? Do you have 'Messy Marvin', a house full of kids, or While natural fibers, like wool, are durable and elegant they are not always the easiest to clean. They are doing some amazing things with synthetic rugs these days. They look great, are very affordable, most are stain resistant and they can be sprayed off with a garden hose! Rug sites, like Rugs Direct, that let you narrow your search by shape, size, fiber, etc. might be helpful with your decorating dilemma. Here is a link that might be useful: Rugs Direct

I think the square one is alright, because it emphasizes the geometry of the table perfectly. A round area rug like the Swirl by EORC adds a real highlight to your room for guests to "marvel at", as it is more unusual especially in greater scale. It is a bit pricy though. But on the other hand it will defo fit in your room, as there are many sizes and colors available. Details and photos here Maybe also suggest to add a painting or picture to the wall, continuing the color palette of the area rug. Do you plan to keep the table and the tufted chairs? cheers

Put your centerpiece on the chess table. Is the other ornate dining set for breakfast, gaming or could it be used here? Do you want to continue with the cow head/skin theme, leaning SW in your decor? They aren't really "cozy and elegant," IMO. Is a new chandy in the picture? That one misses the mark on elegance. You need a minimum of 36-42" from the table top to any baseboards or furniture surrounding the table to allow for chair sit/scoot space. I'd do that measurement first. I really wouldn't want the rug jammed up against the exterior wall. Maybe the smaller, round table would be a better choice for using a rug here.