You get the catenoid by rotating, not the parameterization!
But the original parameterization, (cosh(t), t) is in two dimensions. Which is the "x3-axis"? The only interpretation that makes sense (and gives the correct result) is that we really have (cosh(t), 0, t) in three dimensions.
Okay, rotating the curve around an axis gives a surface which requires two parameters. Since we already have "t", use that as one parameter and, since we are rotating, use the angle of rotation, , as the other. In polar (or cyindrical) coordinates, , . Since we are rotating around the x3 (z) axis, it is the first, x= cosh(t) that is our "r", or distance from the axis. Then and . Of course, z, measured along the axis of rotation, is not changed: z= t. That gives .