Network Activate does pretty much what you thought it does: It activates the plugin on all sites in the network.
It also takes it out of the default Plugin view on the plugins screen for all those sites, so they no longer have the option to either activate it or deactivate it.
So if you’re still seeing it on one of those sites, then it’s either not network activated, or that particular site isn’t in the same network as the others. I’m thinking that this is the case here, because of this line in your question:
If I go to the Network Admin level for this 3rd site in my “network” I
do have the option to “Network Activate” it.
Why would you have a separate “Network Admin” for this site? There is only one Network Admin, per network. Perhaps you have multiple-networks going on here (yes, multi-network is a thing that exists).
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