I am not an accountant. This is not tax advice. However, I do international consulting and have to deal with this kind of stuff all the time. Basically, for him, he needs an invoice. It should describe what he's getting and how much it costs. Then it's up to him to convince his company to pay the invoice. Whether or not they can claim it is an expense is probably important to the company, so something like a support contract is much more useful than a sticker (I really, can't imagine how to declare a sticker from a non-partner as an expense...)

From your point of view, you have sent and invoice and received funds. You need to pay taxes on those funds. How you do that depends a lot on your tax laws. Where I am (Japan), you can be self employed, in which case it is treated just like extra income (Japan doesn't have multiple categories of income, which is super awesome for getting rid of tax evasion). You can also set up various kinds of companies. These companies can shield you from certain kinds of taxes, but accounting becomes much more difficult. Usually it does not pay to set up a company in order to receive $500.

Depending on where you live, you might also be liable for VAT (or the equivalent). Exported goods are usually exempt, but exported services often aren't! Be careful! The reason for this is that value added taxes are usually only taxed once. So if you have a physical good that you are exporting, you have already paid the VAT on the things that made that physical good. Since a service is not made up of physical things, then you are liable for tax on the whole thing.

As is relevant to the original article, where you pay the tax is dependent upon where the company is base, or where the work is done. It gets really complicated from there, but basically if you are providing a service remotely (and not needing a work visa to provide the service), then you are performing the work in your country (even if you are working over the internet) -- or at least that's what my accountant has decided.

TL;DR: giving him an invoice (especially for a service) will make it easy for him to pay you, but will make things complicated for your taxes. It only makes sense if you think you will make a lot of money.