Is there a local variant of TSP?
This is an "Orienteering problem" and is NP-hard [1]. If the starting vertex is given the problem is rooted; your variant is unrooted. Some authors call it the "selective travelling salesman" [2].

[1] B. L. Golden, L. Levy, and R. Vohra. "The orienteering problem". In: Naval Research Logistics 34.3 (1987), pp. 307-­318.

[2] M. Gendreau, G. Laporte, and F. Semet. "A tabu search heuristic for the undirected selective travelling salesman problem". In: European Journal of Operational Research (1998), pp. 539­-545.