Yes, we're all freelancers. We are independent workers. We report to no one (and no, you don't report to clients - but more on that another time) and we govern our own employment. We have autonomy and agency in our careers.
But the word freelancer, along with the words owner, founder, and self-employed, don't truly carry the weight of what you do or they aren't necessarily accurate or reflective of the scope of your job.
Freelancer sounds flimsy, like you might not actually be working, or like you're unsure of who you are and what you do. It also sounds like you might be just starting out and eating Spaghettios for dinner every night. Same goes for self-employed. These terms not only sound vague but also like a deflection to cover up a lack of success. This is not the impression you want to give your friends, your family, and especially potential clients. And everyone you talk to is a potential client.
Owner, on the other hand, carries legitimacy but seems to be reserved for/more applicable to actual brick and mortar establishments that can speak for themselves. I own a restaurant. I own a store. I own a hotel. These are things people can see, understand, and respect.
Founder, like the term CEO, implies a huge significance. The term is often found among big non-profits or VC backed start ups that are quickly making a footprint. Be a founder of Facebook, don't be a founder of Just Starting Out, Inc. Without reputation, calling yourself a founder or a CEO seems like a deception.
Call yourself by your actual occupation. After all, that is what you do. When people ask me what I do, I confidently declare myself a web developer. This declaration provides insight into how I spend my time, and it also demonstrates a highly sought after skill. People don't have to awkwardly fill in the gaps about whether or not I'm truly gainfully employed, and that makes me appear more trustworthy, capable, and professional. At that point the conversation often turns to "Oh wow, you know how to do that stuff? My company/sister/son/bowling league needs some work done. Could you help us?" Yes, yes I can.
Your occupation carries with it an authority and a sense of officialness that isn't always conveyed by the term freelancer or self-employed. And authority and thought leadership are a huge part of not only someone's first impression of you but of their decision to hire you.
Hiring decisions are based entirely on a potential client's perception of you based on what you project through your communication, your portfolio, your attitude, and your confidence. A client's initial perception of you will carry through your entire working relationship and influence how they feel about the work you produce - a confident hire leads to a high quality working relationship which in turn leads to positive recommendations. Don't get started on the wrong foot, and don't let the seeds of doubt be planted in the very first word.
Also, the next time a nagging friend or family member decides to ask when you're getting a real job you can confidently tell them you have one: you're a web developer.