According to The Bookman’s Glossary (R. R. Bowker, 4th edition 1961) Lettre de forme, lettre de somme, lettre de bâtarde are “the three general classifications of Gothic type forms as found in the 15th century. The first is the Pointed and most formal; the second is the Round and less formal; the third is a Cursive form. They correspond to similar classifications of lettering used in the manuscripts that preceded printing.” Nomenclature is of course confusing with something as old as this: lots of people referred to the same thing by different names, and some minority usages still survive in unexpected corners. Just choosing to privilege the French terminology represents a choice of road taken.
In the examples below “Pointed” is referred to as Textura.

From The Dawn of Western Printing at Incunabula.
Here, from Incunabula, is a full discussion of the three forms. It tells us that Gutenberg’s type was based on the script textura quadrata, commonly used for bibles and derived from the “protogothic” script which developed in northern France around the eleventh century. A more rounded script, called rotunda, evolved in Bologna in the 12th century. Lettre bâtarde derived from less formal writing and tended to be used for documents in the vernacular, not in Latin.
Of course it was all more complicated than this, but we can think of scribes using half uncial, chancery hand, and insular scripts, which began to seem unreadable to many as different regions of Europe favored one version over others. Different scripts would be used for different types of book. Alcuin of York (c.730-804) succeeded in persuading scribes to use the more elegant Carolingian minuscule for most purposes, thus allowing international comprehensibility. While the Pointed, Textura style was the basis for the type used by Gutenberg in his Bible, the Carolingian minuscule, a few years later formed the basis for the types which we recognize today as Roman which were introduced by printers in Italy.

Simplified relationship between various scripts, showing the development of Uncial from Roman and the Greek Uncial. From Wikipedia.
Any account of these matters must inevitable over-simplify. Researchers look back on what was the individual practice of thousands of scribes and printers, and impose a pattern on their myriad activities. Such a pattern does provide a sort of history, but doesn’t of course imply any intentionality on the part of the actors. It’s not like Gutenberg said to himself, let’s copy the pointed script. He just knew that this was how you wrote bibles, so, obviously, that’s how you’d print them too. No doubt Aldus Manutius didn’t sit there musing “Why do we have to copy this barbaric Gothic nonsense? Let’s go back to Alcuin and start anew”. He followed the style of earlier classical and secular works, formalizing things as printed types will inevitably do.
My post on Black letter may have some relevance in this context.