Reaction of vinylacetylene (but-1-en-3-yne) with HCl
Beyond a shadow of a doubt, the second explanation, involving an allene rearrangement is the correct one. As you said, alkenes are more reactive towards electrophilic addition and hence the addition of HCl happens first on the double bond to give 3-chloro-but-1-yne. What follows next is an intramolecular rearrangement, through a cyclic five-membered transition state, which is catalysed by CuCl with NH4Cl. It encompasses the abstraction of a hydrogen from the C4, the migration of the chloride from C3 to C2, following the formation of a double bond at C3 and C4. The product is referred to as chloroprene, which gives neoprene on polymerisation, a useful polymer. However the other explanation offered by your facilitator is used by many as a trick to remember the final product, (including, me!). Hope that helps!