Definitely not a fan of reusable cloths or towels as a general rule including in general maintenance applications.
As a minimum if you must use terry towels or washable cloths in your process I would want to see colour coded in a similar structure to brooms, brush ware and other sanitation equipment to segregate from floor use only only through to equipment cleaning. The other issue with towels being washed by a third party provider is how do you really know that they have been washed and sanitised to an appropriate quality level to ensure that they are also microbiological safe.
In our case we use a range of single use wipes (Tork brand) depending on our end use requirement such as simply cleaning spills to wash down of equipment. The ones we specifically use are food approved and low linting and either white or blue colours available from our supplier. To help save on costs we use the available dispensers so that the rolls don't get wet or contaminated and portion control.