If only I could show you a picture... I'll try with words.
When you substitute the position adjacent the exo-methyl group, you generate TWO chiral centers... one at the chlorine position and the other at the methyl position. Before the substitution was made there were no chiral centers. Since you have two chiral centers, you can have both enantiomers and diastereomers. The cis/trans nomenclature refers to the position of the chlorine with respect to the methyl group when viewed along the somewhat puckered plane of the cyclobutyl ring. If both the chlorine and methyl are on the same side of the ring, that's cis. The opposite is true for trans. It really helps to build a model to see this.
When you substitute chlorine at the 3 position of the cyclobutane group, you don't generate any chirality (because, in this case, it is a meso compound) but the chlorine is either on the same side of the ring as the methyl or opposite, hence cis and trans.
In this contex, cis and trans do not refer to any particular mechanism (addition or substitution, for example) for the addition of chlorine. Rather, it refers to the position of the chlorine and the methyl group with respect to a particular face of the cyclobutyl group. Both on the same side is cis. One on each face is trans.
