seishi 精子 is sperm
seiki 精液 is semen
There is no confusion, except common people usually don't make a big deal about it like you do. In the vid cover you put up, the theme is to make the school girl pregnant. If the sex doesn't make pregnant, say she is on the pill, or she has IVD... Or more interesting case, the stud had vasectomy so he shoot semen but it contains no sperm... well that's the case that, scientifically, makes it meaningful to distinguish sperm from semen. But you don't get the opposite case, unless you are doing a procedure in a IVF lab instead of making semen in a vagina. You don't have sperm without the semen. So it's a distinction that makes a difference only to you, no one else.
I think vimana is trying to say something about a 4-kanas word fits more nicely in that sentence than a 3-kanas word. It's a very very Japanese (the language) thing that we have no hope to understand. Anyway I think it may not be as deep as that.
So anyway... I am guessing there exist more vulgar/rude Japanese words for sperm, similar to cum, jizz, spooge, or spunk. That you can't find them in a dictionary.