Electrocautery or lasers will inevitably generate surgical smoke, which hinders the visual guidance of laparoscopic videos for surgical procedures. The surgical smoke can be classified into different types based on its motion patterns, leading to distinctive spatio-temporal characteristics across smoky laparoscopic videos. However, existing desmoking methods fail to account for such smoke-type-specific distinctions. Therefore, we propose the first Smoke-Type-Aware Laparoscopic Video Desmoking Network (STANet) by introducing two smoke types: Diffusion Smoke and Ambient Smoke. Specifically, a smoke mask segmentation sub-network is designed to jointly conduct smoke mask and smoke type predictions based on the attention-weighted mask aggregation, while a smokeless video reconstruction sub-network is proposed to perform specially desmoking on smoky features guided by two types of smoke mask. To address the entanglement challenges of two smoke types, we further embed a coarse-to-fine disentanglement module into the mask segmentation sub-network, which yields more accurate disentangled masks through the smoke-type-aware cross attention between non-entangled and entangled regions. In addition, we also construct the first large-scale synthetic video desmoking dataset with smoke type annotations. Extensive experiments demonstrate that our method not only outperforms state-of-the-art approaches in quality evaluations, but also exhibits superior generalization across multiple downstream surgical tasks.