I live in France and, since I provided my VAT number, have been receiving invoices without VAT, mentioning in the title that this is a tax exemption, but in the small prints of the VAT, it's written that it's Reverse Charged and that I have to take care of paying these charges. Which is also confusing because of that page: https://helpx.adobe.com/fr/x-productkb/policy-pricing/adobe-stores-online-order-payment-faq.html saying that if you live in the EU and provide your VAT number, you won't pay the VAT (which is different from not having the VAT on the invoice and then having to pay it yourself).
I'm not sure what to do, what should I declare, and do I have to pay these taxes?
That is standard how VAT works. The VAT is basically paid only once locally by the final customer. So if you are a business, you have a VAT number and when you buy goods and services from outside of your country of residence, you are entitled to get an exemption, ie buying the goods and services without VAT. With your VAT declaration, you sum up the VAT paid and the VAT recieved from your customer, which is in France 21% if I'm correct, and the difference needs to be paid or gets refunded. Your accountant should know the exact procedure for this.
ABAMBO | Hard- and Software Engineer | Photographer