Before we gave a fast generalized minimum distance (GMD) decoding algorithm for one-point algebraic-geometry (AG) codes. In this paper, we propose another fast GMD decoding algorithm for these codes, where the present method includes an erasure deletion procedure while the past one uses an erasure addition procedure. Both methods find a minimal polynomial set of a given syndrome array, which is a candidate for an erasure-and-error locator polynomial set constrained with an erasure locator set of each size. Although both erasure addition and deletion GMD decoding algorithms have been established for one-dimensional algebraic codes such as RS codes, nothing but the erasure addition GMD decoding algorithm for multidimensional algebraic codes such as one-point AG codes have been given. The present erasure deletion GMD decoding algorithm is based on the Berlekamp-Massey-Sakata (BMS) algorithm from the standpoint of constrained multidimensional shift register synthesis. It is expected that both our past and present methods play a joint role in decoding for one-point AG codes up to the error correction bound.