I know the term "torus" is generally used to describe a circle rotated about some axis that is coplanar with the circle.
Does the the term also apply if you rotate a small section of a circle instead?
Say, take a circle, and then intersect the circle with a line that intersects the circle in two points and doesn't pass through its center. This line will split the circle into a smaller and larger piece. Take the smaller piece and rotate it about the line used to cut the circle. Can that shape correctly be called a torus? If not, what is the correct term?
Usually, the word "torus" implies that the surface is actually homeomorphic to the canonical "donut" shape.
The Wikipedia article mention the names "horn torus" and "spindle torus" for the shapes you get by rotating an entire circle around a tangent or a chord, respectively -- but these shapes would not normally be spoken about as unqualified "tori".
Your construction where a circular arc of less $180^\circ$ is rotated about its chord would produce the central spindle in a spindle torus. Topologically it's just a distorted sphere. You could call it a "spindle", or a "cigar shape", I suppose.