There are many ways of dividing languages from dialects. The three general methods are:
The traditional method has tended to utilize structural and sometimes historical, but intelligibility is also often used. For an example of historical, let us look at some lects in France and Spain.
The various “patois” of French, incorrectly called dialects of French, are more properly called the langues d’oil. It is often said that they are not dialtects of French for historical reasons. Each of the major langues d’oil, instead of breaking off from French Proper (really the Parisien langue d’oil) had a separate genesis.
This is what happened. France was originally Celtic speaking. Around 700-800, the Celtic languages began being replaced by vulgar Latin. People didn’t travel around in those days, so a separate form of vulgar Latin + Celtic evolved in each region of France: Gallo and Angevin in the northwest, Poitevin and Saintongeais in the west, Norman and Picard in the north, Champenois, Franche-Compte and Lorrain in the east, Berrichon, Tourangeau and Orleanais in the center. None of these split off from French (Parisien)!
Each one of them evolved independently straight up from vulgar Latin on top of a Celtic base in their region from 700-1200 or so. The distance between the langues d’oil and French is almost as deep as between English and Frisian.
After French was made the official language of France in 1539, the langues d’oil came under French influence, but that was just borrowing, not genetics.
In addition, in Spain, there are various languages that are not historically related to Spanish. Aragonese is straight up from vulgar Latin on a Basque base, later influenced by Mozarabic. Catalan started evolving around 700 or so. Murcian evolved from vulgar Latin later influenced by Mozarabic, Catalan and Aragonese. Extremaduran, Leonese and Asturian also broke off very early. None of these are historically Spanish dialects because none of them broke away from Spanish!
Of course it follows that langues d’oil, Catalan and Aragonese, evolving independently of French and Spanish from 700-1200 to present, will have deep structural differences between themselves and French and Spanish.
So you can see that the historical way of splitting languages ties in well with the structural method. Where languages have a deep historical split and a millenia or so of independent development, it follows logically that some deep structural differences would have evolved in a thousand years or so. So these two methods are really wrapping around each other.
Now we get to intelligibility. Intelligibility actually ties in well to structural analyses. Linguists who say we divide on structure and not on intelligibility are being silly. Where you have deep structural differences between Lect A and Lect B, it logically follows that you have intelligibility problems. Profound structural differences between two lects makes it hard for one to understand the other. The differential structure really gets in the way of understanding. So once again, one method is wrapping around the other.
As we can see, historical, structural and intelligibility analyses of splitting languages all tend to be part of the same process, that is, they are all talking about the same thing. And they will tend to reach similar conclusions when it comes to splitting languages.