Your controller will most likely have a relay inside with the 2 contacts connected to the rear. If you order the optional outputs then you will get them as depicted, the board inside will have the appropriate modifications for them installed instead of the relay.
The SSR driver simply removes the relay and gives you the coil drive from the relay as a 24V drive for most common industrial SSR units, which operate with an input on the control of 3024V at 30mA.
Trigger output will have a small daughter board with an opto triac on it and the associated 3 resistors and a RC snubber, to provide the control side of a SSR inside the controller and drive to use an external BT157 triac as a control device.
The difference in current is that the alarm relays are typically very small, designed for a limited current and a limited operation cycle, as they typically operate once per power cycle of the machine or less to drive a separate alarm or control circuit. The output relay has to be a better larger unit, which has to handle a moderate current to drive a larger contactor ( thus the 3A rating, though you want the contactor it drives to have a 10VA or less coil or a VDR clamp for a coil under 100VA) for a large number of switching cycles, where it might do 5 cycles per minute in heating for a few years in controlling a heater oven. It has to have a a lifetime of at least 1 million mechanical cycles at rated load.
Standard though on all I get off the shelf is a relay output, so either you use an external contactor to drive the heater ( and in setup do limit the cycle time so it does not try to switch once a second, try 10 seconds or so for a heater using a relay) as the built in relay will only handle a heater that draws less than 3A, which would be 500W in countries with 220VAC mains and 300W in the USA.
I have converted units with relay output to drive a SSR, simply by removing the relay and using 2 47R resistors ( they were conveniently to hand, you can use 2 wire links instead) to link the coil connections to the output pads, and then using the SSR externally, and marking the case with the appropriate changed part number with permanent marker.