It's a lot of different terminology but they're mostly saying the same thing. The blastocyst (in humans, the blastula is called the blastocyst) implants into the endometrium. The reason Cliff's later mentions the trophoblast is because the early blastocyst consists of two parts: an outer layer of cells (the trophoblast, which embeds directly into the endometrium) and an inner cell mass.
The corpus luteum is a completely separate structure that is maintained by implantation of a fertilized egg. It's found in the ovary, not embedded in the endometrium.