Time series anomaly detection is widely used in IoT and cyber-physical systems, yet its evaluation remains challenging due to diverse application objectives and heterogeneous metric assumptions. This study introduces a problem-oriented framework that reinterprets existing metrics based on the specific evaluation challenges they are designed to address, rather than their mathematical forms or output structures. We categorize over twenty commonly used metrics into six dimensions: 1) basic accuracy-driven evaluation; 2) timeliness-aware reward mechanisms; 3) tolerance to labeling imprecision; 4) penalties reflecting human-audit cost; 5) robustness against random or inflated scores; and 6) parameter-free comparability for cross-dataset benchmarking. Comprehensive experiments are conducted to examine metric behavior under genuine, random, and oracle detection scenarios. By comparing their resulting score distributions, we quantify each metric's discriminative ability -- its capability to distinguish meaningful detections from random noise. The results show that while most event-level metrics exhibit strong separability, several widely used metrics (e.g., NAB, Point-Adjust) demonstrate limited resistance to random-score inflation. These findings reveal that metric suitability must be inherently task-dependent and aligned with the operational objectives of IoT applications. The proposed framework offers a unified analytical perspective for understanding existing metrics and provides practical guidance for selecting or developing more context-aware, robust, and fair evaluation methodologies for time series anomaly detection.