How come in the paragraph starting ויוציאנו ה' ממצרים… the first time the Hagaddah says ולא ע"י שליח, and the second time ולא השליח?
From memory (my grandchildren at the seder and from various haggador which I do not have with me). Hashem had appointed Moshe Rabbeinu as the shaliach to bring the Bnai Yisrael out of Mitzraim. When a shaliach is appointed, that agent is supposed to act on behalf of the one who appointed him (as Moshe acted in the previous 9 makos). However, in this final plague, Hashem acted explicitly and did not allow Moshe to be his agent, as he acted in the rest of the Exodus and as the leader in the desert. The other phrases also do not have the hei hayediah because Hashem did not appoint any being to act on his behalf.
The initial statement uses ולא ע"י שליח (does not have the hei hayediah) because it is a basic statement of the general concept. as applied to the entire pasuk. That is He did not assign any messenger or agent to any of the actions shown in the pasuk. Only when the individual explanation is being used, does Hashem say, not only was an agent not used, but even the שליח that had already been appointed was not used for this particular action.