Hybridization in the world of duplication-transfer-loss

It seems to me that the study of reticulate evolutionary histories currently boils down to two options:
(1) reconstructing a species "tree" from multiple gene trees using a coalescent model that includes hybridization (either homoploid or polyploid);
(2) reconciling multiple gene trees with a known [sic] species tree using a model that includes gene duplication, loss and transfer (as well as speciation) - a DTL model.

This often leads me to wonder where hybridization fits into option (2) and where gene transfer fits into option (1). They must fit somewhere. For example, Jacox et al. (2016. ecceTERA: comprehensive gene tree-species tree reconciliation using parsimony. Bioinformatics 32: 2056-2058) describe their DTL as:

comprehensive as it includes the following evolutionary events: speciation, speciation-loss (speciation followed by a loss of one gene copy), gene duplication, gene loss, gene transfer and transfer-loss (gene transfer with loss of the original gene) between two sampled species, and gene transfer and transfer-loss from/to an unsampled species (i.e. a species that is not represented in the dataset) to/from a sampled one.

Since the model is "comprehensive", then hybridization must be included. The only parts of the model that include reticulate histories are gene transfer and transfer-loss, so this is where hybridization must be. Possibly, polyploid hybridization is included in "gene transfer" (an increase in the number of gene copies), and homoploid hybridization is included in "transfer-loss" (maintaining the same number of genes).

This seems to be a simple example of the idea that different types of reticulation events cannot be distinguished from each other. Genomic material moves from one place to another in contemporaneous organisms, either sexually (introgression, hybridization) or asexually (lateral gene transfer). There is nothing intrinsic about gene trees to tell us which mechanism is involved in any given reticulation, other than the relative positions of the donor and recipient in the "species tree" and the possibility of time inconsistency.

This leads to the question of why horizontal gene movement is called "transfer" in one model (2) and "hybridization" in the other (1).