I figure, basically, that I should be able to do whatever things I do on my phone without the fan spinning.
Sorry, I think maybe I misunderstood or something similar. I thought we were talking about laptops that use active cooling, not smartphones with passive cooling.
ah "you are using it wrong", that famously helpful reply.
What? No, that's a very misleading framing of my comment...
Of course it matters what you do on a laptop when it comes to how the fans are reacting. Are you really expecting the fan to stay silent and the computer to stay cooled when compiling the Linux kernel for example?
Different workloads have impact of the thermals, it's as simple as that. Using heavy applications will make the laptop behave differently than if you use vim.
What program are you using for writing said emails? That matters more than what user action you're performing.
As a reference, using gmail in firefox makes my X1 very loud too, but I've moved to Thunderbird as of late and it's much more careful about resource usage.