Spin-Transfer Torque Magnetic RAM (STT-MRAM) is known as the most promising replacement for SRAM technology in large Last-Level Caches (LLCs). Despite its high-density, non-volatility, near-zero leakage power, and immunity to radiation as the major advantages, STT-MRAM-based cache suffers from high error rates mainly due to retention failure, read disturbance, and write failure. Existing studies are limited to estimating the rate of only one or two of these error types for STT-MRAM cache. However, the overall vulnerability of STT-MRAM caches, which its estimation is a must to design cost-efficient reliable caches, has not been offered in any of previous studies. In this paper, we propose a system-level framework for reliability exploration and characterization of errors behavior in STT-MRAM caches. To this end, we formulate the cache vulnerability considering the inter-correlation of the error types including all three errors as well as the dependency of error rates to workloads behavior and Process Variations (PVs). Our analysis reveals that STT-MRAM cache vulnerability is highly workload-dependent and varies by orders of magnitude in different cache access patterns. Our analytical study also shows that this vulnerability divergence significantly increases by process variations in STT-MRAM cells. To evaluate the framework, we implement the error types in the gem5 full-system simulator, and the experimental results show that the total error rate in a shared LLC varies by 32.0x for different workloads. A further 6.5x vulnerability variation is observed when considering PVs in the STT-MRAM cells. In addition, the contribution of each error type in total LLC vulnerability highly varies in different cache access patterns and moreover, error rates are differently affected by PVs.