Summary
Lotus Elise with a negative ground.
Radiator fan which is operated by a relay.
An input from the temperature sensor goes to the ECU.
The ECU then connects the
coil of the relay to ground.
This closes the relay
contacts and connects the positive supply to the fan.
A modification has been added by you.
A relay has been put in place that has the coil of the relay connected to ground.
There is a manual switch that connects the positive supply to the other side of the coil.
The relay contacts are connected on one side to ground and on the other to
another point.
The switch operates the relay and the fan runs.
The only explanation I can see for this working is if the "
another point" is actually the connection from the ECU to the coil of the OEM relay that is labelled as 1-13 in the Elise wiring diagram you posted.
OK therefore I can only assume you have given incorrect information about the first modification.
The connection you have drawn in your original post as going from the rad fan to 87 of your relay must actually go from 87 of your relay to the ECU to coil cable.
Let's start from the basics this diagram shows a basic switch circuit and a basic relay circuit.
In the switch circuit closing the switch operates the bulb.
In the relay circuit closing the switch powers the coil of the relay by connecting it to negative (ground).
Powering the coil of the relay operates the relay contacts and connects the positive of the battery to the bulb and the bulb lights.
OK adding some labels to the relay diagram to match what you have in the car.
This is the OEM arrangement.
The temperature sensor is not shown but just causes the ECU to operate the switch shown as "ECU".
Point A is where you said you were connecting your modification, point B is where I think you have actually connected your modification.
The following is the first modification at the top and the timer relay at the bottom.
The top diagram is how I
think you have connected your first modification and,
if you have done it like this, then the bottom diagram is how I
think your timer relay should be connected. (Where lines cross over they are not connected I just did not have enough space to keep them separate).
This accounts for you saying that you are grounding the fan and it operates.
If this is not enough for you to be able to work this out then I cannot give any more information that might help.