This study develops meta-learners for few-shot weakly-supervised segmentation (FWS) to address the challenge of optic disc (OD) and optic cup (OC) segmentation for glaucoma diagnosis with limited labeled fundus images. We significantly improve existing meta-learners by introducing Omni meta-training which balances data usage and diversifies the number of shots. We also develop their efficient versions that reduce computational costs. In addition, we develop sparsification techniques that generate more customizable and representative scribbles and other sparse labels. After evaluating multiple datasets, we find that Omni and efficient versions outperform the original versions, with the best meta-learner being Efficient Omni ProtoSeg (EO-ProtoSeg). It achieves intersection over union (IoU) scores of 88.15% for OD and 71.17% for OC on the REFUGE dataset using just one sparsely labeled image, outperforming few-shot and semi-supervised methods which require more labeled images. Its best performance reaches 86.80% for OD and 71.78%for OC on DRISHTIGS, 88.21% for OD and 73.70% for OC on REFUGE, 80.39% for OD and 52.65% for OC on REFUGE. EO-ProtoSeg is comparable to unsupervised domain adaptation methods yet much lighter with less than two million parameters and does not require any retraining.