Generative models for deepfakes have rapidly evolved from GAN-based architectures to diffusion models and multimodal large language models (MLLMs). However, empirical research on human performance in detecting deepfake face images has not kept pace. It remains unclear if (and how) detection performance varies across generative models and whether model-specific training improves detection more effectively than general-purpose training. In this paper, we propose the \emph{Deepfake Face Artifact Taxonomy} (DFAT), a cross-model taxonomy of human-recognizable visual artifacts in deepfake face images, based on which we build both general-purpose and model-specific training materials. Through an online experiment ($n = 524$), we examined human performance in detecting deepfake face images and evaluated training efficacy across five representative generative models: three GAN-based, one diffusion-based, and one MLLM (GPT-5). Our results show that participants' detection accuracy was strongly model-dependent and that images generated by more recent and advanced models were not necessarily harder to detect. After training, participants achieved significant performance gains across most conditions, and their improvement varied by model. Notably, model-specific training generally yielded greater numerical gains than general-purpose training. In a five-month follow-up test ($n = 342$), we observed no significant decline in detection accuracy and no significant increase in perceived stress. Overall, our findings provide actionable insights for designing effective, sustainable, and targeted training programs to counter the growing threat posed by deepfakes.

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