Cozy to me means you need to add color or Texture.

Cozy to me also means warmth and neither of your choices are warm. That's ok if they signal cozy to you and your family. Monitor and website colors are so unreliable, but I dont really think either rug goes too well with your sectional. It's a lovely space, so I hope you find the perfect rug.

Not knowing any color or style preference but what you've shown, I just picked a few to show how other styles/colors might look. I posted some on another person's thread recently so the site seemed to pick some of the same ones. These are from the Rug Company

I think this is one of a kind so the size isn't right but i thought the colors were very nice:

Is there an accent color(s) that you would like to bring into the space? Right now, everything is very neutral (gray).

Is your house more modern or traditional? Geometrical patterns lean more contemporary (or could be used in a transitional style room) - the second is a bit more traditional than the first one - but not overly traditional (a Persian rug would be more traditional).

Do you have any artwork for the walls? If yes, selecting a color(s) from a painting as a starting point when looking for an area rug would be a good idea.

Adding some color + texture would help to make the space cozy.

Plus, with small kids - accidents happen. It's nice to have a rug that won't show obvious spills of chocolate milk (or whatever). Rugables are nice because they can be washed easily + you can add a thicker pad so they are comfortable on the floor for the little ones.

Textured wool rugs are nice also

Even a less colorful one can be "cozy" when it's a textured wool rug:

Close up photo:

Colorful washable rugs on Amazon:

Adding textured/patterned throw pilIows +soft throw blankets + curtain panels + artwork + a bit of color + textured rug will all help make the room more cozy. The below photo isn't a perfect example - but it was one that I found quickly:

If you took away all of those things in the above living room, it would look very similar to yours. It's all the details that will make it feel cozy. Also, if some wood pieces were added (for example, the end tables - instead of those white ones), the room would look more comfy/cozy/warm.

We really need direction from you in order to help with specific rug suggestions.