That is the correct style box and it appears to be correctly labeled, what you have described would be a set 1201. One of the 1201 sets in my collection features the same dark green cars.
The box should have some cardboard dividers inside to separate the cars and engine from the track. The set should likely have a full circle of track and 2 straight sections.
The box top should likely have had a large label like this
I know you mentioned that the photo of the black engine and green cars that you showed in a previous post was not the set you are looking at, but here is a picture of an early 1201.
I noted that the photo you showed, does not feature any handrails on the motor, but I was unable to read the number on the engine. The 1201s typically had handrails, with 2 different styles being used over the years. The above style is the early style of handrails. The 1270 engine, which you indicate you have, should not feature handrails, but the majority of 1201s I have seen (but not all) feature handrails.