Ambiguous logical relations in folk art

I found this in the shed of my parents, while taking my son to visit his grandparents. It belonged, originally, to my grandmother. There are two distinct ways to interpret the sentence:

‘Without a cat, a home is just a house’

which was probably the intent of the author, and certainly the intent of my cat-loving grandmother and cat-loving parents. I, however, prefer the alternative, where the phrase ‘without a cat’ attaches to ‘house’ rather than the sentence as a whole. :
or, to make it explicit:

‘A home is just (a house without a cat)’

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