Perhaps some more examples will be useful.
Look at Artin, fig. 1.4 on p. 156, depicting a stylized stem with leaves on alternating sides. This figure has glide symmetry, because a reflection in the horizontal line (the "stem") followed by a horizontal translation (through the distance between two consecutive leaves) carries it back to itself. Note that it does not have any reflective symmetry, i.e. there is no line over which reflection will carry the figure back to itself. The point is that, although the glide is a composition of a reflection and a translation, neither of these motions by itself carries the figure back to itself. After the reflection alone, the leaves are on the wrong sides of the stem; same with the translation alone.
Notice the same is true of Isaac's ...bpbpbpbp... example. A reflection in the horizontal axis turns everything upside down. Thus it is not a symmetry of the figure because it changes the figure. (If you blink while I perform the reflection, you will still know something is different.) However a glide, i.e. the composition of this reflection with a horizontal translation the length of one letter, will carry each b to the next p and vice versa. This is a symmetry because it carries the figure back to itself. (If you blink while I perform the glide, you will not notice that the figure has changed.)
One more (beautiful) example: M.C.Escher's Horsemen:
There is a glide (reflection over a vertical axis followed by a vertical translation) that carries the red horsemen to the white ones and vice versa. However, there is no reflection that by itself carries horsemen to horsemen.
As for why there isn't a separate name for a translation composed with a rotation, joriki's answer is the reason: (rotation + translation) is a rotation about a different point. To illustrate, consider rotating the plane 180 degrees about the origin and then translating it 2 units to the right. The rotation sends $(x,y)$ to $(-x,-y)$ and the translation sends this to $(2-x,-y)$. Notice that the composed map $(x,y) \mapsto (2-x,-y)$ is the 180 degree rotation about $(1,0)$.