Almost all modern performances and recordings of natural trumpet music are actually played on instruments which do not really resemble originals from this era. At first, one hole was drilled to correct the F and A by shortening the tube, but then additional holes were introduced to increase the security of players by eliminating the possibilty of sounding certain nearby notes. My belief is that this music should be played with instruments and techniques as close as possible to those used when the music was new. If a musician finds it difficult to do, then he needs to practice harder. The horn is based on an original by Johann Leonhard Ehe III made in 1746.
None is particularly historically accurate.
I also coach the Natural Trumpet Group of the Towson University Early Music Ensemble, which does not use ventholes or tapered leadpipes. Since I'm playing fifth part here, I chose a copy of a 1632 Hanns Hainlein trumpet made by Bob Barclay, crooked to modern C, and played with a Webb 'Harper Sr' mouthpiece. Several years ago, we experimented with using English style slide trumpets. These work similarly to a trombone, in that the tube length is variable in order to allow more notes than available on the standard natural trumpet. However, since the slide moves at the back of the instrument, the mouthpiece needs to be placed at an angle so that the slide does not strike the player's face.