Synthetic data generators and machine learning models can memorize their training data, posing privacy concerns. Membership inference attacks (MIAs) are a standard method of estimating the privacy risk of these systems. The risk of individual records is typically computed by evaluating MIAs in a record-specific privacy game. We analyze the record-specific privacy game commonly used for evaluating attackers under realistic assumptions (the \textit{traditional} game) -- particularly for synthetic tabular data -- and show that it averages a record's privacy risk across datasets. We show this implicitly assumes the dataset a record is part of has no impact on the record's risk, providing a misleading risk estimate when a specific model or synthetic dataset is released. Instead, we propose a novel use of the leave-one-out game, used in existing work exclusively to audit differential privacy guarantees, and call this the \textit{model-seeded} game. We formalize it and show that it provides an accurate estimate of the privacy risk posed by a given adversary for a record in its specific dataset. We instantiate and evaluate the state-of-the-art MIA for synthetic data generators in the traditional and model-seeded privacy games, and show across multiple datasets and models that the two privacy games indeed result in different risk scores, with up to 94\% of high-risk records being overlooked by the traditional game. We further show that records in smaller datasets and models not protected by strong differential privacy guarantees tend to have a larger gap between risk estimates. Taken together, our results show that the model-seeded setup yields a risk estimate specific to a certain model or synthetic dataset released and in line with the standard notion of privacy leakage from prior work, meaningfully different from the dataset-averaged risk provided by the traditional privacy game.