Diffusion models are a new class of generative models that have shown outstanding performance in image generation literature. As a consequence, studies have attempted to apply diffusion models to other tasks, such as speech enhancement. A popular approach in adapting diffusion models to speech enhancement consists in modelling a progressive transformation between the clean and noisy speech signals. However, one popular diffusion model framework previously laid in image generation literature did not account for such a transformation towards the system input, which prevents from relating the existing diffusion-based speech enhancement systems with the aforementioned diffusion model framework. To address this, we extend this framework to account for the progressive transformation between the clean and noisy speech signals. This allows us to apply recent developments from image generation literature, and to systematically investigate design aspects of diffusion models that remain largely unexplored for speech enhancement, such as the neural network preconditioning, the training loss weighting, the stochastic differential equation (SDE), or the amount of stochasticity injected in the reverse process. We show that the performance of previous diffusion-based speech enhancement systems cannot be attributed to the progressive transformation between the clean and noisy speech signals. Moreover, we show that a proper choice of preconditioning, training loss weighting, SDE and sampler allows to outperform a popular diffusion-based speech enhancement system while using fewer sampling steps, thus reducing the computational cost by a factor of four.