I have my views on this but I appreciate others may think differently.
The way I operate is that [nearly] all of my finances are just nebulous, part of the same thing. So I don't have a budget for different things, I just have one generic 'pot' of money even if it is stored in different places. One month I might spent £500 another month I might spend £5k and that money isn't assigned to any purpose until it is spent. To put it another way, the "intended use" for savings is constantly evolving. You never know what the future will bring, you might think that today that money is going towards a holiday but if the washing machine breaks down next week maybe that's no longer the priority for that money.
One of the common fallacies (again, from my personal perspective) I see is people who are taking out debts that are more expensive than funds they already have available. In your case buying a washing machine on credit only makes sense if the interest rate charged is less than you get on savings (unlikely unless you pay it off straight away). The message I struggle to get across to people is that by doing this, they will have LESS savings in the long term, because the cost of servicing that debt will impact on their ability to build savings. So if you buy a washing machine for £500 on savings instead of credit then yes tomorrow you have £500 less savings. But in a couple of years time your savings should be bigger than had you bought the washing machine on credit because you will have spent more than £500 plus savings interest on that washing machine due to the higher interest rates on the debt.
It's a similar equation with people paying for insurance monthly, to me it makes no sense if they can pay it annually and avoid the interest, as after 1 year they will have more money than had they paid it monthly (unless they have a good investment vehicle for the capital, obviously).