You are correct in your question regarding the use of the word game in Richard Connell's short story The Most Dangerous Game. In the text, the use of the word game could be used in either sense of the word.
The problem arises when one looks at the definition of the word game. Game, according to Merriam-Webster, is defined in the following ways:
1. An activity engaged in for diversion or amusement.
When this definition is used, one could only apply it to the text in regards to General Zaroff's "understanding" of the term. Neither for Rainsford nor the others Zaroff has hunted would consider the "activity" a "game." Only for Zaroff would the game be entertaining.
2. A physical or mental competition conducted according to rules with the participants in direct opposition to each other.
With reference to this definition, both Rainsford and Zaroff would be considered playing a game. Unfortunately, Rainsford does not have a choice in the matter. He (Rainsford) must simply adhere to Zaroff's rules (play or die) based upon the fact that he wishes to live.
3. Animals under pursuit or taken in hunting.
Using this definition allows one to use the second suggestion you have made regarding the use of the word "game" in the text. Unfortunately, Zaroff has eliminated the animal and replaced the animal with a human being.
Therefore, the use of the word game in the title of the text could easily be used to define either the competition between two hunters or the quarry which a hunter is chasing. Regardless of the one chosen, either could be easily justified based upon the definition applied to the word game.