Variable name is quiet correct. At least it's very common:

- https://wiki.archlinux.org/index.php/Environment_Variables#Examples
- https://help.ubuntu.com/community/EnvironmentVariables#Preferred_application_variables
- http://askubuntu.com/questions/27447/google-earth-and-browser-environment-variable

Though one would better export it like this:

``` bash
if [ -n "$DISPLAY" ]; then
	export BROWSER=xdg-open
else
	export BROWSER=links
fi
```

I would suggest to output propose showing similar snippet (using `open` when on MacOS and `xdg-open` on Linux as GUI browser).