Does the Mount Rinjani summit and lake trek finish in the same village where it starts?

No, the classic three-day summit and lake trek is a point-to-point route, not a loop. It starts at Sembalun village on the eastern side of the mountain and finishes at Senaru village on the northern side, which means your start point and your exit are about 45 to 60 minutes of driving apart.

The geography explains the design. From Sembalun the trail climbs roughly 9 kilometers of open savanna and rolling hills to Plawangan Sembalun, the crater rim camp at about 2,639 meters. The summit push from that side covers the final approach to the 3,726 meter peak, then the route descends into the caldera to Segara Anak, the crater lake, before climbing out to the Senaru rim camp and dropping through tropical forest to the Senaru trailhead. Walking up the dry eastern flank and down the green northern jungle gives you both faces of the mountain, and no itinerary that returns to its start village can offer that.

The practical consequences are worth planning for. Your vehicle cannot simply wait where you began, so the operator sends a pickup to the exit point. Your clean clothes and travel bag should be waiting at the finish, not the start. And if you left a suitcase at a guesthouse in Sembalun, it will not walk to Senaru by itself.

This is exactly why we base ourselves in Senaru. Our guests sleep at our homestay the night before, we drive them across to Sembalun in about 45 to 60 minutes for registration and the start of the walk, their luggage stays secure in Senaru, and they finish the trek right where a shower, a meal, and their bag are already waiting. If you stage from Sembalun instead, ask your operator in writing how your bag gets to the finish village and who is responsible for it along the way, because there is no standard system for that transfer between villages, only whatever the operator commits to in writing.