What are the different notions of one-way functions?
Usually one-way functions are used for crypto, and so you want that no efficient adversary can invert the function. Identifying efficient adversaries with randomized polynomial-time, you get  the typical notion of security which talks of randomized poly-time machines. But of course you can think of different security notions.

For super-fast crypto, you may want the one-way function to be computable in restricted models. Here a great result by Applebaum, Ishai, and  Kushilevitz shows that a poly-time computable OWF implies a OWF where each output bit depends on just O(1) input bits (which is arguably one of the simplest computational models you can think of).