Is the "Use on Civitai Generation Service" license prohibiting image generation on a local PC?

I've noticed a new commercial use license called "Use on Civitai Generation Service" has been emerging recently.

Does this license prohibit image generation on a PC that I personally own? In other words, does having this license mean that models with this license can only be used to generate images on Civitai and not on a local PC?

No. What you activate is what you authorize in terms of the use of your model. Note that the default is on.

For "Use on Civitai Generation Service" it means that you as a creator authorize the commercial use of your model on Civitai's generation service.