I think a term was defined as one of the elements in a sum, the same way that a factor is defined as one of the elements in a product.
So "transient term" isn't part of standard mathematical terminology?
I would say it's standard, but why search for ''transient term'' and not just ''transient''. It usually is a term when it appears in mathematics, because in the driven harmonic oscillator case, it is the homoegenous (unforced) solution and as we know the solution is biult up as a sum of a homogeneous and an inhomogeneous solution. So there are two parts of the solution, one transient which we see nothing of after a while, and one steady state (particular solution in harmonic oscillator case) which is the only thing we see after a while. Generally it is ''what dies with time'' and I see no need for it to be more precise than that.