"Beer Ireland grants permission to any professional member to use the logo on any of their beers which were brewed in a micro brewery (as defined by the Revenue Commissioners in RoI or HMRC in NI) on the island of Ireland, and on associated marketing and packaging materials."
Don't know what the ICBI terms are, but I'd imagine they amount to the same thing; i.e. if you're drinking a beer with either logo you can be sure it's not "fake craft".
As for the differences between BI and ICBI: BI is for any individual working in the micro beer industry in Ireland, ICBI is for physical breweries (not individuals) in RoI. They're quite different in their roles, not competitive, and there is a lot of overlap.
One interesting detail is that Beer Ireland ratified its constitution earlier this year using the EU recommended limit (200khl) to determine membership eligibility, not the local one. This means anyone is eligible to join as long as they are not working for or associated with a brewery that produces more than 200khl (and other conditions). However the use of the logo is still restricted to the local limit (30khl in RoI, 60khl in NI).
As for two logos, my opinion is that if people can get their heads around that there are close to 100 breweries, probably 500+ beers, they can manage two logos!