Introduction:
The distinction between proterostome and deuterostome embryos is based on the development of the blastopore and the fate of the mesoderm. In proterostomes, the blastopore becomes the mouth, while in deuterostomes, it becomes the anus. The mesoderm in proterostomes arises from a group of cells near the blastopore, while in deuterostomes, it arises from pouches that bud off the archenteron.

Developmental Stages:
The process of embryonic development involves several stages, including fertilization, cleavage, gastrulation, and coelom formation.

Fertilization:
Fertilization is the process by which the sperm and the egg fuse to form a zygote. In both proterostomes and deuterostomes, fertilization occurs externally or internally. In most cases, fertilization occurs externally in proterostomes, such as insects and mollusks, and internally in deuterostomes, such as humans and sea urchins.

Cleavage:
Cleavage is the rapid division of the zygote into smaller cells. In both proterostomes and deuterostomes, cleavage occurs after fertilization. In proterostomes, the cleavage is spiral and determinate, meaning that the fate of the cells is determined early in development. In deuterostomes, the cleavage is radial and indeterminate, meaning that the fate of the cells is not determined until later in development.

Gastrulation:
Gastrulation is the process by which the blastula is transformed into the gastrula, which has three germ layers: the ectoderm, mesoderm, and endoderm. In proterostomes, gastrulation is characterized by the formation of a blastopore, which becomes the mouth. In deuterostomes, the blastopore becomes the anus. The mesoderm in proterostomes arises from a group of cells near the blastopore, while in deuterostomes, it arises from pouches that bud off the archenteron.

Coelom Formation:
Coelom formation is the process by which the body cavity, or coelom, is formed. In proterostomes, the coelom forms by the splitting of the mesoderm. In deuterostomes, the coelom forms from pouches that bud off the archenteron.

Conclusion:
The ability to distinguish a proterostome embryo from a deuterostome embryo first arises during gastrulation, when the fate of the blastopore and the mesoderm becomes apparent. Therefore, the answer to the question is C. Gastrulation.