How do I model my professional status when I work from a home office?

Sometimes I wonder what my kids think of their remote-working mother. They must know I work—I feel like I spend 90% of their summer vacation ordering them out of my home office—and yet they will bypass their dad to interrupt me because he’s “busy.” Is this because of deep-seated gendered biases that they have absorbed from our culture? Is it because they think I’m a softer mark? It’s hard to tell.
Remote work has given me more freedom to choose the type of mother I want to be. Before my daughter was born I decided I would be the mother who cooks. I am also the mother who works full time, but that was a given. I prefer to keep my family in rent and groceries. Ergo, I work full time. Cooking from scratch is my choice. I grew up in a family that showed love through good food. As an adult I equate simple ingredients with health. And so I make time before work, on my lunch breaks, and after work, to do things like make bread or simmer chili. Instead of commuting to work I make waffles.
I love that I can be the mother I want to be. I also worry that my kids think I’m doing it all. Or that they discount the outside work I do because I perform it inside the home. My daughter in particular is trying to show maturity by taking on the domestic tasks that I normally perform. She spent this weekend making salsa and baking bread. She also tries to soothe her little brother when he gets hurt and help him with his homework. I am both flattered that she wants to emulate me and worried that she assumes this is what women do.
I want both of my children to learn to cook good food. I consider it a life skill. But I also want them to know that they can choose what they’re known for. Cooking does not define me, but it is part of the complicated definition of me. I cook because I want to, because I enjoy it, and because I think it’s important, not because it’s expected of me. And so I talk about why I cook, and I also talk about the things I do for work.

I think some of the meta message must be getting through. Grandma recently sent both kids a sum of money. My daughter approached me this weekend with a plan to use that money to start a baking business. Did she approach me because I’m the softer mark? Or did she correctly identify the parent with the business background? It’s hard to tell. All I know is that it’s her choice, and I’ll be supporting her startup any way I can.