First decide how you want to use your spaces. You don't have to blindly follow the labels on the realtor floor plan.

In a house my parents owned years ago, they used what was called the family room on the plan, as a dining room. In my previous house we used the "formal living room" as the main TV watching area and made the "family room" as the more formal entertaining space.

Where do you want to sit with family, with visitors, when entertaining large groups?

Do you have kids?

Will you have one main dining area or two?

Do you have a backyard? Will you entertain there? How do you get to it?

For best help with layout you will be asked for all these dimensions ...

Thanks so much for your answer! The part of the blue square is 9 (going horizontal) x 13( going vertical). That is all the space to the right of the entryway. I do

If if place a rug horizontally like I do now, part of the rug ends up showing up/cutting through half of the entryway door like in the first picture. I can't decide if I need a bigger rug going horizontally or keep the 8x10 going vertically.

The "entryway" should not drive your layout design for the whole space. It doesn't matter that you see rug/furniture when you come in the door, as long as the furniture isn't right up against the hallway and blocking flow through the house.

Add all the dimensions if you want some of the Pros to do some furniture layouts for you.

10’ x 14’ is a standard size, and would work better than 9’ x 13’ here.

Test drove the 10x14 by marking some edges. Does this look better? Is it ok if the rug goes over the narrow entry way (as you can see in the second picture) and goes a little into the "office" area. The bench will become a desk and small office area.

The third and 4th picture shows how the 9x12 rug will look, marked in blue tape for the narrower 9 feet

There are two distinct areas. In the blue, turn the rug. Doing so frees up the foyer floor entry into the room. Having the rug intrude into another area looks rediculous.

Two chairs in front of window.

Move the t v (Install a swivel arm) to the other wall (where the box is) with a slim console under.

Now for the other area.

Keeping the 3' walk way from the entry, you furnish the fireplace area. I'd float a sofa or sectional (sized for the space) to the fireplace. The blue area chairs/windows in the background.

For the two window area, add a game table.

I wouldn’t buy the rug yet. Since you haven’t chosen the 3-seat sofa or finalized the furniture layout, I’d solve the circulation first and let the rug follow the seating arrangement — not the other way around.

What bothers me more than the actual rug size in the photo is that it visually projects into the doorway. With this room connecting the entry, kitchen and bedroom, I’d keep those circulation paths very clear and create a distinct seating “island” inside them.

Once the sofa and any chairs are positioned, I’d choose the largest rug that comfortably anchors that grouping without entering the main walkways. In a room this large, 9' x 12' may ultimately look better proportionally than 8' x 10', but only if the furniture placement supports it. Ideally, at least the front legs of the sofa and chairs should sit on the rug.

Since you’re still building the room from the ground up, it may also help to decide what you want the finished space to feel like before buying the major pieces. I recently transformed the same living room four completely different ways, changing the furniture, lighting, colors and overall atmosphere. It might give you some ideas for the direction you want to take yours:
https://youtu.be/R-2ZyyBc6EY