No virus protection in the BIOS.
Took some more time doing a reinstall, and the "boot = sda2" line isn't the critical option, it definitely seems to be the active flag that makes the difference.
boot = /dev/sda
bitmap = /boot/slack.bmp
bmp-colors = 255,0,255,0,255,0
bmp-table = 60,6,1,16
bmp-timer = 65,27,0,255
timeout = 50
vga = normal
image = /boot/vmlinuz
root = /dev/sda2
label = linux
Above works as long as partition 2 is set active. Turn the flag off, no boot.
My PC (Asus A8N5X motherboard) running Slack 13.37 also had the second partition set active (the first partition contained an XP install). Turning that flag off, made no difference: both Slack and Windows continue to start up normally on that machine.
So I think now it must be some BIOS peculiarity on the laptop.
And of course, I've now found someone who had a similar problem:
Thanks for the help, guys.