I think either could be the right choice.
There are a number of jobs where you pay someone for completing a particular task and they are only paid when that happens. You are not their employer as such but someone who hires them to do a task for a small number of hours. Examples might be weekly such as gardeners, childminders, dog walkers, piano teachers etc.
These people get paid for the hours they work. There is no sick pay or holiday pay from you as a user of them.....it is up to them to add into their fees something to cover this stuff.
If they don't come to work to do their job because they are sick or because something else prevents them coming and doing it like weather, you don't have to pay them. They are not entitled to pay.
So for example, when school was closed and the piano teacher couldn't teach, they didn't get paid. When Gardner wasn't able to do his work because of snow, he won't have been paid.
It seems harsh, but these workers are not entitled to be paid for work not done, unlike employees who may find their work place shuts for the day.
What usually happens to make this a better outcome for cleaners, piano teachers, gardeners etc is that they and the people they work for try to arrange an alternative slot - so the work still gets done and they still get paid.
If it's not possible to arrange an alternative slot, if you use one of these workers it is up to you if you choose to pay them.....but they cannot expect it. You wouldn't pay a cleaner who didn't come due to illness or due to their car breaking down. You wouldn't pay the window cleaner who can't come because his van isn't working. You wouldn't pay a piano teacher who had to take a day off to look after a sick child.
If you - the person who hires them decides they can't come because it's inconvenient to you, then with enough notice it should be possible to re-range, but if you are cancelling them short notice, I think payment is then due.
Odd circumstances such as weather preventing work - something outside everyone's control are unusual - and something that the self employed person isn't covered for and has to work around. Most contractors or temporary hourly paid staff won't get paid if the weather prevents them getting in. An employer might choose to pay them for an odd day - that's fine.....but you also have to be aware of setting a precedent.
Best, as always to clarify exactly when you will and won't pay with these kinds of work. The general rule is that payment is due if work is carried out or if the person who hires them cancels with short notice - short notice being defined.