The panino caprese has one structural enemy that the insalata caprese never faces: the bread. On a plate, the water that fresh mozzarella and ripe tomato shed runs harmlessly to the edge. Trapped between two slices, that same liquid soaks straight into the crumb, and within minutes a sandwich that started crisp is a sodden one. This is the defining problem of the caprese as a sandwich, and every good version is really a set of answers to it. The flavours are settled, tomato, mozzarella, basil, oil; the engineering question is how to keep them from destroying the thing that holds them.
The craft is moisture management from the very start. The tomato is sliced and often salted and drained, or laid on paper, so it gives up its water before it ever meets the bread rather than into it; the mozzarella is drained and patted, because a wet ball of fior di latte is mostly liquid waiting to escape. The bread is chosen with a tight, sturdy crumb and a real crust, a ciabatta or a firm roll, since an open soft bread surrenders immediately. Oil is brushed on the cut faces as much as added for flavour, glazing the crumb so it resists the wet rather than drinking it. And the sandwich is built to be eaten soon, because there is no construction that survives a caprese sitting for an hour. The whole skill is in buying time.
The variations are mostly different solutions to the same leak: the lightly grilled or pressed version where heat firms the bread and drives off some water, the one built on a denser focaccia that can take more moisture before it fails, the caprese that swaps wet fresh mozzarella for a firmer, drier cheese to cut the problem at the source. Each of those is a distinct fix worth its own treatment, and each deserves its own article rather than being crowded in here.