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.
This ignores how much more wasteful with resources developers are on the desktop, though.
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.