I would move the TV away from the windows and put it where the desk is. Create your sitting area around that, which would still incorporate the window, but not have the TV back-lit. Put the desk or a built-in desk on the slopped wall where the horses are. You could do more built-in storage where the big sectional is and make it a kid's corner.
https://www.houzz.com/photos/2013-parade-of-homes-dream-home-edina-mn-traditional-family-room-minneapolis-phvw-vp~3210248
https://www.houzz.com/photos/2013-parade-of-homes-dream-home-edina-mn-traditional-family-room-minneapolis-phvw-vp~3210250

I'd make it very kid-friendly, fun and light-hearted as these two ideas are. On your picture #1, build in a computer desk right of the partition, moving the TV to the current desk wall. Then divvy the rest of it up by activity; music, play, exercise? etc. as needed. As the kids grow replace play area with air hockey area.

Would you still recommend putting the tv where the desk is even though it's on a partial wall? See image below. Thanks again for the advice and pictures!!

I don't think it makes a difference that it's a partial wall. It will be a dedicated space for the TV which I think is good. It's not a huge long wall that you'll have to figure out how to decorate.

If the dimensions permit I would create an office area in the niche that is to the right of the family area, that will create more of an "area" or space if you will. Just a thought...

Our house is kind of a funny house, probably more house than we need. So we already have a dedicated room for an office, a library and a craft room for the kids, in addition to the usual bedrooms and living room. That's why I'm so confused with what to do with this very large room (it's around 600 sq ft). Maybe we need to get a couple of game tables and make it more like a rec room? I appreciate everyones' suggestions, just still feeling stuck…. Should I try to separate the area that now houses the tv from the area around the keyboard/sectional/horses area?

You can separate the area with the placement of the larger furniture pieces without enclosing anything. Game tables would be great, you have the room.

How about putting the TV under the sloped roof (to the right of the arched window) and pull the corner sectional away from the wall, turn it around so that one part faces the TV and the other part faces the window in that part. So it creates a 'closed off' TV corner (does that make sense?). You can put one of the chairs under that window angled towards the TV maybe.

Then place the leather couch under the feature window and create a separate seating area there with the leather furniture. It will make space for a games table in the centre of the room.

Piano where the desk is.

I like that carolins. It used to be like that with the tv corner. The sectional is not terribly comfortable, and the brown leather chair is my husband's tv chair, so I wonder if it would look weird to do a tv corner but with chairs and the couch instead…wouldn't make as much of a barrier. I may give it a try, thank goodness for the furniture moving disks!

hahaha... definitely worth a try. And since you have the furniture moving disks... shuffle away :-) Sofa facing TV and leather chair with its back towards the open space?