So, there's actually a difference between "addressing the senate" and what we're talking about. Have you ever gone to a meeting that employed robert's rules? I'm assuming you have (the senate, obviously, doesn't, but the concept is similar). The guy that adjudicates all the nonsense about points of order is the presiding officer. They're not formally addressing the meeting when making those decisions.
The senate is seriously a body that lacks a ton of formal rules. They're not like the house. And yeah, the vice president is the guy who formally has the job of presiding officer (though they can also delegate that authority IIRC). Here's a
link that talks about the powers of the majority leader.