To me they are analogous.
I think switching up the metaphor could help. To me, a DM is the architect. An architect's job is to design a space that serves the people who will use it. An architect who designs a playground does want to entertain the kids who will use it.
The playground architect isn't necessarily pushing the kids on the swing or playing lava tag. But they are, in a way, entertaining the kids by designing a space in which the kids can have fun.
I see my role as a DM as being a playground architect. I am designing structures through which my players create fun. At the same time I can be a really adaptive architect and add, subtract, or change my structures as needed to make things even more fun for the players.