Short video has witnessed rapid growth in the past few years in multimedia platforms. To ensure the freshness of the videos, platforms receive a large number of user-uploaded videos every day, making collaborative filtering-based recommender methods suffer from the item cold-start problem (e.g., the new-coming videos are difficult to compete with existing videos). Consequently, increasing efforts tackle the cold-start issue from the content perspective, focusing on modeling the multi-modal preferences of users, a fair way to compete with new-coming and existing videos. However, recent studies ignore the existing gap between multi-modal embedding extraction and user interest modeling as well as the discrepant intensities of user preferences for different modalities. In this paper, we propose M3CSR, a multi-modal modeling framework for cold-start short video recommendation. Specifically, we preprocess content-oriented multi-modal features for items and obtain trainable category IDs by performing clustering. In each modality, we combine modality-specific cluster ID embedding and the mapped original modality feature as modality-specific representation of the item to address the gap. Meanwhile, M3CSR measures the user modality-specific intensity based on the correlation between modality-specific interest and behavioral interest and employs pairwise loss to further decouple user multi-modal interests. Extensive experiments on four real-world datasets demonstrate the superiority of our proposed model. The framework has been deployed on a billion-user scale short video application and has shown improvements in various commercial metrics within cold-start scenarios.