Your problem is the +12v that is present on pin E of the relay isn't making it to the fuel pump. If that voltage made it to the pump, your pump would run and your problem would be fixed.

A few things. Test with a 12 v trouble light. Using a meter puts no load on the circuit and can give you misleading information. Test with everything plugged together. If you had the 2 sec +12 on E with the fuel pump unplugged, reconnect it and test again.

Your problem is not the ignition switch.

If indeed you have +12v on E of the relay for two seconds but nothing at the pump you will either have to find why it's not connecting, find the break in your wire, or add an additional wire between the two. Most likely your problem is a connector that is corroded and not making contact. Is the point of your tester piercing the corrosion but the terminals are not making contact? Use the point of your tester to poke directly to each sides terminal and verify the voltage is making it across. Also look to make sure the wire itself is not corroded off of the metal terminal. Again this is where the test light comes in play. You might have a high resistance connection that with no load allows the +12 to cross, but with load the voltage drops too low. The test light puts some load on the circuit and can show the high resistance joint. A meter is designed to put practically zero load on the circuit.

Many of the relays used in 88 were of poor design and if left dangling up side down by their wires filled up with water and corroded the relay socket terminals badly.