Opinions please. When end user is not paying for service, who is my customer? For example public facing government project

My last job was this exact scenario. Our product was free for end users but paid for by website owners who wanted it for their visitors. It's a very difficult model and you need to get pricing and features right.

The people who pay are your customer. It's partly about positioning yourself as experts in your field to let you say "we know what users want and you'll get more/have less support/empower yours if you buy our software".

There is a maximum that says "when you do not know what the product is, is that the product is you", even if customers do not pay, they are still customers, the product will be polished as they request services, until they find the product they could pay for