No, push button to ignition key changed much earlier. The change to the ballast resistor happened in 1970, well over half way into Series 2. The relay is still a starter relay but of a different type to cope with the feeding power from the main circuit for cranking and via the ballast resistor for running. See previous post at the top of this page.
A different coil is indeed required.
For cars without the ballast resistor it is part C30120 Lucas HA12 (45208A). 12 volt coil.
Car with the ballast resistor uses C32579. Superseded by C35812 Lucas 16C6 (45227). This is a 6 volt coil.
Ignition switch was also changed at this point, from C.28522 to C.35846
This also coincides with a number of other changes: sump; fuel tank (unvented); heater and choke with illumination.
From
http://www.xke-lovers.com/ballast.htm
The purpose of this change, according to service bulletin No. 1 B 11 was to provide maximum spark efficiency at high engine speeds and to facilitate engine starting in very cold conditions.
A new ignition coil is used in ballasted applications, Lucas 16C6. This ignition coil has a primary resistance of 1.43 - 1.58 ohms. It is used with a ballast resistor, part number 3BR, with a resistance of 1.3 - 1.4 ohms. Because the ignition coil and resistor have nearly equal resistance, under normal conditions about one half of the nominally 12 volt battery voltage appears across the ignition coil. During starting, voltage drops, therefore it is necessary to bypass the resistor while the starter is cranking.
To achieve this, the regular starter relay has been replaced by a special relay, part number 33231. This relay has two separate contacts, C1 and C4 that are connected to the armature contact, C2 when power is applied to the relay coil through W1 via ignition switch contact 3. The W2 winding of the relay coil connects to frame ground to complete the circuit. Relay contact C2 is directly connected to the battery side of the key switch circuit, supplying nominal 12 volts at all times.
Contact C4 of the relay connects to the solenoid to provide voltage to operate it when the relay is energized. Contact C1 connects between the ignition coil (+) terminal and the ballast resistor, supplying the nominal 12 volts directly to the ignition coil and effectively bypassing the ballast resistor while the starter is energized. Once the key is moved back to the normal run position, voltage is removed from the relay coil causing contacts C1 and C4 to open which removes voltage from the starter solenoid and the junction of the ignition coil and resistor.
Since the run side of the key switch (terminal 2) is now connected to the tach, ballast resistor, ignition coil and distributor points, the car continues to run although the coil voltage is now approximately 1/2 of what it was during the starting sequence.
Series 2 factory fit...?