How can an Arduino determine the status of a washer or dryer using only vibration data?
I think the simplest option is to experiment first: record the vibration
level as a function of time during one wash cycle, and do that with a
few different washer models. Then look at your data and devise an
algorithm with your eyes on the measured curves. This is really the
only reasonable way to do any kind of signal processing: look at the
signal first. If you still have problems finding an algorithm, you can
come back here and show your data.

Unless you live in a region with huge geologic activity, chances are
that the vibration level of any washing machine will be well above the
background, which should allow you to choose a single vibration
threshold suitable for every model.