Personal property is private property.

The term "personal property" is mostly used by Marxists.

That is not correct, the term is used by all kinds of socialists. You may not care about the difference in general, but denying it is not helpful when discussing socialism. You're trying to refute the claim that private property (in the socialist sense) is problematic by defending private property (in the liberal sense), which doesn't make sense because these are two different concepts.

Before I address your point, I'll clarify that I want to use correct/standard definitions, not definitions only understood by small subsets of the population.

Now to your point: private property intended for personal use, which socialist ideology claims is not problematic, is the direct progenitor of private property used for other purposes, like earning an income for the owner from being rented out, or from being utilized in an enterprise that employs other people to operate the private property.

The differences between private property intended for personal use, and private property intended for business or commercial use, are completely superficial with respect to the basic principles that society operates on.