Wow! I see that this is a deeper and more correct way of using the store's features, but this option can not be implemented in the Starter version. Thank you anyway!
Jerome wrote: Since you can use discount to be applied on categories ; you can have create categories for your different discounts and use the parent category (named for example "discounts") to apply your badge on all.
I imagined such a structure of the store and I had a question - how can you manage the term of the discount with this structure? After all, with this structure, to give a discount, you need to assign a category to the product, for example, 5% discount category. In this case, even if the discount can specify the validity period, then after the end of this period the products will not disappear from the category itself, so the visitor will see the badge SALE, but there will be no discounts for the products. Probably, I do not know something, but it looks like such a structure is inconvenient for managing discounts. Although it is convenient to assign the SALE icon for different discounts, but if they are pre-defined and tied to a specific category.
In the end, this is not the idea that I meant. I spoke only about one badge SALE, which will always be for any product that has any discount regardless of the size of the discount, and the date it was created. And that it works always after a one-time setup.