My understanding is that the cephalochordates are named as such (meaning “head chords”) because the notochord reaches into the head. From Wiktionary:
Any of the primitive fishlike creatures, of the subphylum Cephalochordata, including the lancelets, that lack a true spine but have a notochord which, in the case of cephalochordates, reaches into the head, contrary to the Urochordata.
Urochordata, by contrast, means "tail chord", though I cannot find a reputable source (or any, really) that describes or details why this is the case.
I can surmise it's simply because the notochord doesn't reach the head of a tunicate embryo -- as is corroborated by an image from Lemaire 20111 copied below.
Image D' shows notochord in red. Credit: Lemaire 2011
However, I would prefer a quotable, reputable source that indicates explicitly that the Urochordates are in fact named as such due to their notochord not reaching their head.
If this is not the case, then my question more broadly stands at: Why are the Urochordates named the way they are?
1: Lemaire, P., 2011. Evolutionary crossroads in developmental biology: the tunicates. Development, 138(11), pp.2143-2152.