If the tanks are side by side, the water level in the big tank is higher than the small one, This would cause the small tank to constantly over flow as it tries to reach the level of the big tank water line. It would do this unless both water lines where already level, which is very rare, unlikely.
Did plumber take cylinder away? if so, can you get it back?. very easy to test for hole. one enter and one exit, put cylinder on side , block one hole with hane and blow through other, you will hear and feel pressure escaping if indeed there is a hole.
We can assume the leak is quite big, since water is filling tank all the time and we can also assume it is on the ground floor, because if it wasn't you would have wet patches on ceilings.
I have mixed feelings about leak sealing agents, as sometimes they can cause more harm than good, however I have used an agent recently that claims to seal leaks of up to 200L a day. I 'll give it a week and retest system to see how it goes. It came from Sweden and is quite expensive (€60). It is thinner(like water) than most sealers on the Irish market.