It's obviously a method that transforms a PDE into an infinite system of ODEs.

And even if it's possible to solve the ODEs analytically, it's not, in genral, possible to sum the series analytically.

However, it's advantage over finite difference methods for solving PDEs is that it describes derivatives much more accurately. Therefore a finite sum approximation is much more accurate than the result of a finite difference approximation.

It is a variable reduction method (PDEs into ODEs) widely useful for subsequent numerical analyses

From your description, the procedure you describe looks very much like what is usually called the pseudo-spectral method, see e.g.

Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (2007). Spectral methods: fundamentals in single domains. Springer Science & Business Media

Fornberg, B., & Sloan, D. M. (1994). A review of pseudospectral methods for solving partial differential equations. Acta numerica, 3, 203-267.

In the general nonlinear case, it is a numerical method, not a semi-analytic one. If you look at the space discretization only what you call 'reduction method' is often named method of lines in the numerical literature. In the linear case, it is a semi-analytic method in the sense that for example for any linear constant coefficient pde with periodic boundary conditions you can have a representation of the exact solution, which can then be computed (in principle!) to arbitrary accuracy.

https://en.wikipedia.org/wiki/Partial_differential_equation