The lobster grinder is defined by the bread it is built on, and that bread is not the split-top frankfurter roll the New England lobster roll requires. A grinder runs on a long sub roll with a real crust and an airy interior, the same loaf that carries an Italian cold-cut grinder, and that single substitution changes the whole sandwich. The split-top roll exists to be a soft, griddled spine for a few ounces of barely-dressed meat; the grinder roll is a structural shell built to hold a long, heavy, loaded sandwich. Putting lobster into it makes a different food: not the restrained two-bite roll of a seafood shack but a full-length sub that happens to be filled with shellfish.
The craft is in scaling lobster to a sub rather than to a hot-dog bun. The meat is picked into large pieces, bound in mayonnaise, and built down the length of the roll over a bed of shredded lettuce with sliced tomato, the same architecture as any cold grinder, so every bite holds lettuce, tomato, and lobster together rather than lobster alone. The lettuce is doing real structural work here that it never does in a Maine roll: it lines the crumb against the wet salad and adds the cold crunch a long, soft, mayonnaise-bound filling has none of, and the tomato brings the acid. The roll's crust is the reason the format holds, since a foot of dressed lobster salad would collapse a split-top but a sub roll carries it. This is the lunch-counter and sub-shop reading of lobster, sold by the inch off the same line as the meatball and the Italian, generous where the shack version is precise.
The variations are mostly the regional argument the grinder steps outside of. The Maine lobster roll dresses the meat lightly and serves it cold on a buttered split-top; the Connecticut lobster roll serves it warm in drawn butter on the same roll. The grinder takes neither side and answers a different question, how to make lobster a full sub rather than a delicate roll. A hot version warms the meat in butter before it goes on the sub roll, edging back toward the Connecticut idea on grinder bread. Each of those is a codified build with its own defenders and deserves its own article rather than being crowded in here.