Monoclinic and triclinic
Ivan (in the comments) is spot on about these two. Among all the systems, these are the only ones with angles that are not fixed by symmetry, so you actually have to measure them.
Rhombohedral or Trigonal
These are two different settings of the same lattice. The unit cell can either be a prism (with angles of 90, 90 and 120 degrees, blue lines in the figure) or a rhombus (with all three angles equal, and all three sides equal, black lines in the figure).
In general, there are infinite choices for unit cells (you just need three translation vectors that are not linearly dependent, so pick an origin on a lattice point and three other lattice points and its a unit cell). The conventional choices usually have more of the symmetry elements along the axes, face diagonal or body diagonal, and keep the volume of the unit cell fairly small.
This used to be called pseudo-orthorhombic.
Again, you have a choice of unit cell, and the more uncommon one is a rhombohedral prism. When you switch from all right angles to a rhobohdral prism setting, face-centered turns into primitive and primitive turns into face-centered.
Here is a quote from Wikipedia:
In the orthorhombic system there is a rarely used second choice of crystal axes that results in a unit cell with the shape of a right rhombic prism;2 it can be constructed because the rectangular two-dimensional base layer can also be described with rhombic axes. In this axis setting, the primitive and base-centered lattices swap in centering type, while the same thing happens with the body-centered and face-centered lattices.
Historically, the first data available to crystallographers (and minerologists) were the shapes of crystals. Orthorhombic crystals can have faces that are at 90 degree angles or not, depending on the kinetics of growth. Pictured below are three crystal forms with underlying orthorhombic symmetry, with the 2nd one showing is a shape with (+/- 1, +/- 1, +/-1) faces labeled orthorhombic pyramid:
Cubic, Hexagonal and Tetragonal
See OP's comments for those.
The German Wikipedia article on crystal systems has the terms Bravais used in 1886:
- Assemblages terquaternaires (cubic)
- Assemblages sénaires (hexagonal)
- Assemblages quaternaires (tetragonal)
- Assemblages ternaires (trigonal)
- Assemblages terbinaires (orthorhombic)
- Assemblages binaires (monoclinic)
- Assemblages asymétriques (triclinic)