Can an organism obtaining a part of its genome via horizontal gene transfer be called a "hybrid"?
As you point out, there isn't a singularly accepted biological defnition of 'hybrid.' The most basic would be a single organism that exhibits traits of two individual organisms, but that will find perfectly acceptable disagreement depending on who you talk to. You, for instance, are human - but a hybrid of your parents' genomes. 

So I would personally say that horizontal gene transfer is equivalent to gene inheritance, and that as long as the organism exhibited traits of both 'parent' individuals - then yes, it is a hybrid. Some might include that even if it didn't exhibit traits and merely contained the genes in any inherited genomic data that the hybrid label would be appropriate.