Reliable human-machine discrimination is becoming increasingly important as large language models and autonomous agents are deployed in online settings. Existing approaches evaluate whether a system can produce behavior or responses indistinguishable from those of a human, following the emphasis on outputs as a criterion for intelligence proposed by Alan Turing. Cognitive science offers an alternative perspective: evaluating the process by which behavior is produced. To test whether cognitive processes can reliably distinguish humans from machines, we introduce CogCAPTCHA30, a battery of 30 cognitive tasks designed to elicit diagnostic process-level features even when task performance is matched. Across the battery, process-level features provide stronger discriminative signal than performance metrics alone, reliably distinguishing humans from agents even under output matching (mean process-feature classifier AUC = 0.88). To evaluate agentic process differences, we compare off-the-shelf frontier agents (Claude Sonnet 4.5, GPT-5, Gemini 2.5 Pro), Centaur (a language model fine-tuned on 10.7M human decisions), and two task-specific fine-tuning approaches applied to Qwen2.5-1.5B-Instruct: action-level supervised fine-tuning (A-SFT) and process-level fine-tuning (P-SFT), which directly optimizes process features. Broad fine-tuning on human decisions improves human-like task processes relative to off-the-shelf agents, while task-specific process-level supervision further improves behavioral mimicry. However, this advantage diminishes under cross-task transfer when supervised process targets do not naturally generalize across tasks. Explicit process-level supervision can improve human behavioral mimicry, but only if appropriate task-specific process representations are available, highlighting process specification as a bottleneck for achieving human-like cognitive processes in machines.