Virus-to-virus gene transfer via "sequences integrated into a common host organism"
Does it mean that a viral gene or other DNA sequence gets integrated
  into the host's genome and then some other virus (that chanced to be
  in the same organism) manages to obtain this sequence and package it
  into its own DNA?


Yup, that's what it means (although not all viral genomes are DNA-based).

A lot of genomes contain extensive content that is thought to arrive from viral transfer. Here's a reference to get you started:


Endogenous Viral Elements in Animal Genomes (by Aris Katzourakis , Robert J. Gifford; 2010; PLoS Genetics) 


If much of an organism's genome originally derives from viral transfer, then any subsequent viral transduction represents a virus-to-virus transfer of genetic information.