"Generative AI" is a misnomer; it's not the same kind of "generative" as the G in GAN.

While you're right about GANs, diffusion models as transformers as transformers are most commonly trained with supervised learning.

I disagree. Diffusion models are trained to generate the probability distribution of their training dataset, like other generative models (GAN, VAE, etc). The fact that the architecture is a Transformer (or a CNN with attention like in Stable Diffusion) is orthogonal to the generative vs discriminative divide.

Unsupervised is a confusing term as there is always an underlying loss being optimized and working as a supervision signal, even for good old kmeans. But generative models are generally considered to be part of unsupervised methods.