We say batting cages for the facility, and tunnel for the 'lane.' When you have coin-op (some are even outdoors!), we just say cages. When they are on the side of the field, we say cage.
Same here. Our local place has coin op cages and then individual tunnels that are numbered. The tunnels can be adjusted by sliding the nets back and forth to create larger or smaller spaces, so you can rent tunnel 1, or 1 & 2 and so on.