I use both, depending on what I am trying to do. Sometimes both at the same time...
The ethernet adapter in the IO tree is for the Redlion to be the EIP slave (implicit). I use this, for instance, if the Redlion G3 is a data concentrator to the Logix platform. It makes it easy to get data from any protocol the G3 supports into the Logix system. This method makes the G3 a slave and the Logix master reads data at the RPI interval set up in RS5000. Gateway blocks are required in G3 here to send data back and forth. However, I think RS5000 complains if we have more than 125 registers configured, and it can only handle one connection to the G3 by way of IP address. Even though C2 allows up to 4 ethernet protocols - only one per logix system seems to be allowed per AB limitation. C3 allows more that 4 ethernet protocols.
If the G3 is just an HMI, I would use one of the tag based drivers - native tag or l5k. This makes the G3 the master and it pulls data from the Logix platform. No gateway blocks are generally required here, the tags have all the mapping info required.
Sometimes (most of the time!) the G3 panel is doing both jobs - so I might use both drivers as a way to 'logically' separate the data.
In either case, these connections would count against your limits. I have installed several similar systems to yours and never had any trouble with number of connections that you mention. The AB documentation has a nice worksheet on estimating EIP capacity but I haven't done a system big enough where I needed it yet so I can't drop the particulars from memory. Other on the forum will know exactly the CIP limitations and could help you more if the docs are confusing.