Universal Ignition Switch

[Tifosi]

I suppose if you wanted to play diagnostics games, you could unhook the ignition switch entirely and use toggle switches to run the engine (one for 'key on' and one to operate the starter) and see if the pumps still did what they do at the same regularity as they do now.

If so, then the relay is indeed the answer.

Or, next time you turn the key off and the pumps are still running, give the panel a stout thump to see if that stops them.

Again, if so then the relay is the answer.

[papajam]

The fuel pump circuit is designed/wired such that the fuel pump will receive power from the fuel pump relay only if;
A) the engine is cranking or
B) the engine is running

So even with the key on and engine off, the fuel pump does not receive power (assuming everything is working as it should ). This eliminates the ignition switch.
If you'd like a fuel pump relay for testing purposes, let me know and I'll send you one.