I think, Semi analytical method is attached to solving PDEs especially when the model is linear while approximate analytical methods are used for solving ODEs. In summary, methods such as DTM, HPM, VIM, VPM, DJM, ADM etc when used to handle ODEs generate approximate solutions.

Approximate analytical method

DTM is a numerical method for solving differential equations. The concept was first introduced by J.K. Zhou in "Differential Transformation and Its Applications for Electrical Circuits", 1986. and it was applied therein to solve both linear and nonlinear initial value problems in electric circuit analysis.

Dtm in my opinion is a semi exact method of analytical solution. It is an alternative procedure or technique for deriving analytic Taylor series equation of differential equations. Which may be applied directly to nonlinear ode/pdes without discretization. Thank you sir.