Deep cycling refers to allowing the battery to run all the way down before recharging. It is best to only let it go about 3/4 of the way down before putting it on the charger. It is not like the old cell phones that had memory, so you wanted to run them all the way down. These batteries will last longer if you don't allow them to completely discharge.
Slim line charger is the newer chargers that Vapor4Life sells with its products. It is a smart charger, so it knows when to stop charging when the battery is ready to go. It also adds to the life of the battery. It is the only type charger, Vapor4Life currently sells, they discountined the other type about a year ago, so if you have received a new charger in the last year, more likely is is a slim line charger.