OSi Developer Guide
MapGenie is a brand name. From a developers point of view, MapGenie represents a number of tiled, cached map services. These are token secured. The token is specific to the server on which your application is hosted and has a fixed lifespan.
MapGenie services are primarily available in ITM, with only some available in Web Mercator and IG projections
In all my samples I have used the Web Mercator based services. My reason was down to expediency (laziness). I had the lat/long coordinates for the Phoenix Park - where the OSi are based - and couldn't be bothered looking up the ITM equivalents. As it happens, most Leaflet and OpenLayers samples use Web Mercator.
For further details see the OSi's MapGenie page.