Yes, you have missed something.
No, do not run Disk Utilities from an earlier version of OS X
The Snow Leopard disk has a copy of Disk Utilities on it.
Once you have booted from the disk, start to move through the install process, but only one step at a time.
When you reach the right place, you will see the menu bar at the top of the screen.
One of the options is called Utilities, or words to that effect, and contain things like the Password Reset Utility and Disk Utility.
Launch from there and you are good to go
I have recently tried to run Onyx which has identified a problem which requires me to run disc utilities from the start up disc.
The Snow leopard disc I have only appears to offer Install option.
Q1 Have I missed something?
Q2 Otherwise should I try to run Disc Utilities from an earlier version of OSX