Why does it imply that? Many homeless have alternatives, but they aren't either applicable, or the person don't simply want that. Just as one example, a homeless person with a dog could probably get rid of their dog so they can stay at the homeless-shelter, but instead chose their close bond with their dog over that. Does that suddenly mean the person isn't homeless?

Another (personal example) is when I first arrived to Barcelona and barely could afford food. I spent two nights sleeping outside in the city instead of paying hostel fees, so technically I had the choice of spending a bit of money so I had roof for the night. Lets say that situation was longer instead of just two nights, would I not count as homeless then because I could have spent my money differently?