What's it called when a male seahorse gets "inseminated"?
Short answer
I think that "egg deposition" by means of the female "ovipositor" seems the most commonly used terminology. 

Background
A cursory Google search yielded quite a few alternatives, ordered here from the most fancy to the least:


...that are deposited directly into the brood pouch ... (Look et al., 2007)
...transferring their eggs directly into the male brood pouch... (Masonjones &amp; Lewis, 1997)
...the female injects her eggs into the male’s pouch using her penis-like ovipositor
...once all the eggs are inserted


References
- Look et al., J Exp Biol (2007); 210: 432-7
- Masonjones &amp; Lewis, Copeia 1996; (3): 634-40