Deep convolution neural networks have been widely used in medical image analysis, such as lesion identification in whole-slide images, cancer detection, and cell segmentation, etc. However, it is often inevitable that researchers try their best to refine annotations so as to enhance the model performance, especially for cell segmentation task. Weakly supervised learning can greatly reduce the workload of annotations, while there is still a huge performance gap between the weakly and fully supervised learning approaches. In this work, we propose a weakly-supervised cell segmentation method, namely Multi-Task Cell Segmentation Network (MTCSNet), for multi-modal medical images, including pathological, brightfield, fluorescent, phase-contrast and differential interference contrast images. MTCSNet is learnt in a single-stage training manner, where only two annotated points for each cell provide supervision information, and the first one is the centroid, the second one is its boundary. Additionally, five auxiliary tasks are elaborately designed to train the network, including two pixel-level classifications, a pixel-level regression, a local temperature scaling and an instance-level distance regression task, which is proposed to regress the distances between the cell centroid and its boundaries in eight orientations. The experimental results indicate that our method outperforms all state-of-the-art weakly-supervised cell segmentation approaches on public multi-modal medical image datasets. The promising performance also shows that a single-stage learning with two-point labeling approach are sufficient for cell segmentation, instead of fine contour delineation. The codes are available at: https://github.com/binging512/MTCSNet.