Automated lumbar spine segmentation is very crucial for modern diagnosis systems. In this study, we introduce a novel machine-agnostic approach for segmenting lumbar vertebrae and intervertebral discs from MRI images, employing a cascaded model that synergizes an ROI detection and a Self-organized Operational Neural Network (Self-ONN)-based encoder-decoder network for segmentation. Addressing the challenge of diverse MRI modalities, our methodology capitalizes on a unique dataset comprising images from 12 scanners and 34 subjects, enhanced through strategic preprocessing and data augmentation techniques. The YOLOv8 medium model excels in ROI extraction, achieving an excellent performance of 0.916 mAP score. Significantly, our Self-ONN-based model, combined with a DenseNet121 encoder, demonstrates excellent performance in lumbar vertebrae and IVD segmentation with a mean Intersection over Union (IoU) of 83.66%, a sensitivity of 91.44%, and Dice Similarity Coefficient (DSC) of 91.03%, as validated through rigorous 10-fold cross-validation. This study not only showcases an effective approach to MRI segmentation in spine-related disorders but also sets the stage for future advancements in automated diagnostic tools, emphasizing the need for further dataset expansion and model refinement for broader clinical applicability.