The tuna salad sandwich is the plain tuna mayo sandwich after the salad vegetables have been folded in, and the defining decision is that they go into the bowl, not on top as a garnish. Drained tinned tuna is bound with mayonnaise and then mixed through with chopped salad vegetables, cucumber, tomato, sometimes spring onion or shredded lettuce, before any of it meets the bread. The vegetables are not a layer the way they would be in a club; they are dispersed through the filling so each forkful of the mixture carries both fish and crunch. That integration is the whole point of the form and the thing that separates it from a tuna sandwich with a few slices of cucumber laid in.
The craft is moisture management against a filling that has gained water. Tomato and cucumber both weep, and a salad mix that sits even a short while releases liquid that thins the mayonnaise and runs into the crumb. The standard answers are to deseed the tomato and salt and drain the cucumber before they go in, to keep the mayonnaise slightly tighter than a plain tuna sandwich would take so it can absorb the extra water without sliding, and to assemble close to when the sandwich is eaten rather than letting it stand. The vegetables also have to be cut small enough to distribute evenly, because a large slab of tomato in a bound filling shifts and slides where a fine dice stays put. The bread is soft and plain and buttered to the edges, the butter doing double duty here as flavour and as a moisture barrier against a filling that is wetter than the baseline by design.
The variations move along the salad rather than away from the fish. Tuna with sweetcorn folded through is the lunchbox standard and arguably the most common reading of the form; a tuna crunch leans hard on cucumber, pepper, and onion for a filling that is mostly texture; a fuller salad build adds leaf and tomato until it approaches a composed sandwich. The hot version sealed under melted cheese, and the egg-and-olive Niçoise reading, are separate sandwiches entirely. Each deserves its own article rather than being crowded in here.