Thanks.
Actually I've just remembered that I have some old Irish postcards and one of them, dated Jan 1912, is from a New Zealand visitor to Belfast, writing home.
It says:
"After a very smooth passage down the river Clyde (full moon) we are in Belfast at 6:30 am. I've rode in an Irish Jaunting Car to the hotel....."
Maybe that's what the occupation was then, he owned his own car and used it as a taxi and to take tourists around?
The postcard sender also mentions going to a performance of 'The Wearin' O the Green' at the Theatre Royal - packed house, had to stand.
I've found the father also quoted on his daughter's wedding registration (in 1919) as 'Car Driver' (he was deceased by then, having died in 1898).
Edit: just had a look again at that postcard and I see that the sender has put an 'X' beside a little horse and cart in the foreground, indicating the type she means in the message.
I've found similar car's in other postcards and the rider seems to be sitting on the side of the car, not at the front. Perhaps that's just the poor image quality or maybe he was just resting on the side while waiting for a fare.