So funny - very similar! In the am while I'm having a coffee she plays and I sit on the couch and check email etc (computer is tucked on a small table out of reach otherwise she'd get at it!). She'll come and go, maybe bring stuff to me, and I'll give her a bit of attention and she'll walk away. Often I also sit on the floor and we'll read, sing interactive songs, cuddles, peekaboo (I'll go around the house with this and she'll follow me). If she knows I'm accessible (i.e dining room, kitchen) then she tries to engage me; if I have to go somewhere off-limits she plays a bit on her own but eventually comes to find me...
Other than going out, I'll bring her up to my room to goof around on our bed. She has a few baskets of toys, so I keep some in each room in case she wanders. Recently I bought a wooden tray toy so just before a meal I sit her into her chair and she plays with it.
I guess we're all intuitively doing similar stuff so it's gotta be somewhat right. I talk to her a lot, naming things etc. So I'm sure they're just soaking it all in. I have noticed though that when she's well fed and rested, she tends to play on her own longer, and some days she seems more independent and others not. Depends on her mood I guess!