Multi-interest recommendation is essential for the matching stage. By generating multiple user representations, it can better cover the diverse interests derived from user interaction history. Ideally, multi-interest models should effectively identify the underlying latent factors — the specific themes, intents, or preferences — within historical behaviors. However, conventional methods typically rely on weighted aggregation (e.g., Attention), which we argue leads to over-smoothed representations. This aggregation dilutes the intensity of significant patterns that appear only locally, blending them into a blurry average. To address this, we propose DMI, a model-agnostic diffusion framework that distills precise interests by amplifying co-occurring latent factors across behaviors. Distinct from prior diffusion works that reconstruct the single next item—which risks collapsing diverse interests—DMI reconstructs the interest vectors themselves to preserve their distributional independence. To support this, we introduce a cross-transformer module that adaptively extracts interest-specific information from designated historical interactions, transforming the diffusion process from an unconditional one into a guided, interest-disentangled pathway. In addition, we design a gradient back-propagation strategy to decouple the joint optimization of the reconstruction and recommendation losses, thereby improving training stability. Extensive offline experiments demonstrate DMI's superiority over existing methods, achieving an average relative improvement of 11.2% across all metrics on Amazon Books datasets while increasing recommendation diversity by 11.8%. Successfully deployed in a real-world recommender system, DMI effectively enhances user satisfaction and system performance at scale, serving the major traffic of hundreds of millions of daily active users.