Is there a set in stone distinction between the meaning of the phrases "meeting someone" and "have a meeting with someone"?
Yes. In the phrase "have a meeting", a meeting is an arranged event at which a group of people come together to discuss a particular topic. The phrase is most commonly used to refer to business meetings, at which a group of employees discuss, for example, the work they intend to do on a particular project. At a political meeting, members of a political organization would get together to discuss the party's policies and so on.
"To meet somebody" just means to come into their presence, either accidentally or deliberately. "Let's meet at the railway station at 7pm" means "Let's both go to the railway station at 7pm and find each other." Presumably, we'd then catch a train together to go somewhere, or maybe it's just that the station is a convenient location to find each other and then we'll go do something else. In contrast, "Let's have a meeting at the railway station at 7pm" would be rather strange – it would be an invitation to something like a business meeting but in a very unusual location for that.
"Meet" can also be unplanned. "I met John in the park" could just mean that I'd gone to the park and I happened to find John there. Or it could mean that the first time I ever spoke to John was in the park. If you wanted to emphasize that, you could say "I first met John in the park."
It is clear to me that the first implies a first meeting and the second does not.
"Meet" doesn't have to be for the first time, as the "Let's meet at the railway station" example shows. And one could meet somebody for the first time in a meeting: for example, it would be very common for employees of a company and customers to meet each other for the first time at business meetings.