Although the terms "natural trumpet" and "Baroque trumpet" are often used interchangeably when discussing trumpets from the 17th and 18th centuries, both terms can take on a variety of meanings. To understand how they are used today, and how they relate to the two major schools of Baroque trumpet performance practice, it's helpful to consider some historical context.
The Natural Trumpet
Technically speaking, the term "natural trumpet" can be applied to any lip-blown instrument of a fixed length—without valves, slides, or vents—that can play only the naturally occurring notes of the harmonic series. In context of early music performance, the term denotes the twice-folded trumpets that emerged at the end of the 14th century in Europe. This basic form remained relatively unchanged for 400 years, covering the Renaissance, Baroque, and Classical periods of European music.
The Baroque Trumpet
In the recent past, the term "Baroque trumpet" has been used to refer to any trumpet that plays Baroque period repertoire. Even the modern valved piccolo trumpet has sometimes been referred to as a Baroque trumpet, but its length, bell profile, mouthpiece size, bore size, sound quality, and manner of playing couldn't be more different than a historical trumpet. Since the start of the early music performance practice movement (from the middle of the 20th century), the term "Baroque trumpet" has been increasingly used to describe replicas of historical trumpets with the addition of vent systems. These were invented in the middle of the 20th century for tempering the 11th and 13th partials (those that sound out of tune in context of equal temperment) and to provide more security and facility. It's important to note that experts now generally agree that trumpeters in the Baroque period did not use vent systems, rather they tempered the "out of tune" notes through their choice of equipment, manner of playing, and musical approach.