Whatever you decide to do, I would, personally, want the toys as far away from the stairs as possible for safety reasons, and tucked into that lower left corner to keep them out of sight when the area gets a bit messy. :-) Of the layouts proposed by ER, my favorite is in the latest batch of 4 (upper left quadrant), right above my post, except I'd put the toys where the exercise equipment is, and have the exercise stuff by the stairs/bathroom wall. I especially like the symmetry of the TV/furniture arrangement with the windows.

Since it is going to be a play area I would go for the vinyl. Although carpet is warmer, play dough and paint is easer to clean off of vinyl. You can always add a area rug for floor play.

Not sure why pressure from other family who aren't living in your house would make you rethink what you want in a house if it wasn't on your radar to begin with. I agree with patricia about having living spaces on your main level be living spaces, and having a separate living space without a TV so adults can separate from kids but still be on the same level. I would make the office space your TV room (den). With doors. Kids toys could be in there too. Regarding the office, what does your upstairs look like? How many bedrooms do you have? If you have 4, you can easily convert one bedroom into an office. Most people respond to that by saying "but I NEED a guest room!" If that's your response, think about how often you have overnight guests, and whether you want to dedicate 144 square feet of your house to guests who will be with you a handful of nights each year (when you think about your house as a cost/SF, think about how much money you're spending for that dedicated space, and how many nights in a hotel it would take to recoup). I am planning on converting our guest room into a play room/guest room, with a murphy bed in the closet for the handful of times we have overnight guests each year. You could do the same with a guest room/office combo. Murphy bed or pull out couch or day bed for overnight guests.

Congratulations. It looks like a wonderful home to raise your family. The most important thing is that they are playing happily and safely where you can easily interact with them and supervise them. This is their home and they are doing the important and wonderful work of play. So for these ages the basement does not work. A 2 year old (and even the 4 year old) need to be on the same floor level as the supervising adult, and your main floor room is ideal for that purpose. I would not close off the room too much. Maybe a curtain across the opening is all that you need. You need to hear what's going on. Or sometimes an unnatural silence is a clue that you need to pop your head in to investigate! Paint one wall with blackboard paint - it comes in many colours! There are lots of ideas on the internet for organizing smaller items (Duplo, books, puppets, stuffed animals/dolls, pretend hats/clothes). My granddaughters (ages 2 and 6) are often here and I have their things in heavy canvas bags with handles, the kind that you'd put beach stuff in. Some bags go in open cubbies and some into an armoire. It's easy to dump the bag somewhere to play, and then remind the children that one bags gets filled up and put away before a different bag gets dumped. https://mnmomma.com/ikea-toy-storage-hacks/ I love this idea! https://www.amazon.ca/Large-Children-Football-Organizer-Storage/dp/B01JLVC7NG/ref=dp_prsubs_1?pd_rd_i=B01JLVC7NG&psc=1 Or this one https://www.walmart.ca/en/ip/Toy-Storage-Organizer-Play-Mat-Outdoor-Basket-Large-Drawstring-Portable-Container-Bins-Floor-Activity-Kids-Room-Classroom-Gray/33XMDXW9HR61 I've seen deep old TV armoires repurposed very attractively as children's play places with lots of storage. https://thethriftycouple.com/5-ideas-to-repurpose-furniture-into-creative-play-for-kids/  Things like riding cars etc are more in-your-face but no less important for a child's development. This solution would work indoors too - a table with a curtain as the garage door. https://ourhousenowahome.com/2016/07/hide-kids-outdoor-toys-diy-storage-solution.html Enjoy that toys are part of your life now and don't stress that your house doesn't look like a sterile furniture store advertisement. If I walked into your house and the first thing I saw was your children's play area I'd think "Wow, what a wonderful family to grow up in!"