I guess there are many local differences. The first 30 years of my life I lived in Graz, and then another 20 years in Vienna (both in the eastern part of Austria), and I don't feel comfortable with this terms:
- die Wäsche einfüllen
- die Wäsche in die Maschine tun
- die Wäsche in die Maschine machen
- die Wäsche in die Maschine packen
- die Maschine einräumen
The term that I use is:
die Wäsche in die Maschine geben
I will explain why I don't feel comfortable with the other terms:
When I hear this word, I always have an association with »eingießen« (english: to pour), so my first thought when I hear this verb, is a liquid. Someone can einfüllen a detergent or any other liquid into a machine, or you can even einfüllen sand, or bigger amounts (1000+) of tiny particles, so that you also can pour them like a liquid.
I know, that einfüllen also has the meaning of put something into something else, so that it is correct to say »Ich fülle die Wäsche in die Maschine ein«, but I feel not comfortable with this construction.
This simply sounds like poor German to me. I know, there are regions in Germany, where is is usual to say »Kannst du bitte die Teller auf den Tisch tun«, but where I live (and lived) this sounds like the German of someone who is learning German as a foreign language and who don't remember the correct verb.
The phrase »in die Maschine machen« (without explicitly saying what you put into the machine) has a very dirty meaning. I means »to shit into the machine«. When you add »die Wäsche«, you make indubitably clear, that you mean »to put the laundry into the machine«, but a small part of each Austrian German native speaker's brain comes up with the other meaning for a fraction of a second. Maybe for such a short time, that it won't bubble up to awareness, but still enough to produce a strange vibration.
Packen as used in this phrase »die Wäsche in die Maschine packen« is not Austrian German. This is not how packen is used in Austrian German. It is German German. But since we in Austria are so used to German German, nobody would say it is wrong, but no native speaker of Austrian German would ever use this construction. Ok is »etwas packen«, for example: »einen Koffer packen« (to pack a suitcase«), but not ok is: »Etwas irgendwo hin(ein) packen« (»Ein Buch in den Koffer packen«, »to put a book into a suitcase«).
For me this verb means »to clean up by putting things into the container where they belong to«. When I was a child, my parents told me thousands of times: »Räum endlich dein Spielzeug ein!« (»Put your toys back to the boxes where they belong!«) So »einräumen« has to me the meaning of »create order«, so is something that must be done painstakingly. But that doesn't match with the process of putting laundry into a washing machine, because it doesn't matter how the socks and shirts are laying in the machine after you finished einräumen.
You can einräumen a dishwasher (German: »Geschirrspüler«, but very often also called just »Maschine«). Because here it is important that the plates are positioned correctly, and that knifes, forks and spoons are in their box.
But again I want to make clear, that this all is true for Austria (at least for the eastern part of this country). Maybe there are different usages in other regions for all verbs discussed here.