Programmer: dismissive term used by people that don't really understand programming. You will often encounter it in industries where software is just one of the many supports for the main product. The pay is usually low. The people doing the hiring don't really care what you do or what your skills are. They are just using a checklist of buzzwords and standard tests found on the internet. The work environment is either bad or boring. And coworkers are usually not interested in evolving professionally or personally.

Developer: term used by software people. The kind of persons that are plugged into the tech and IT world one way or another. They usually understand how a product is built and the efforts involved. Because of this, they are often prepared to pay for your work and invest in your skills. The hiring process usually involves higher level concepts and is done in a more sociable manner. The work environment is mostly good, but overworking, missed deadline anxiety and crisis management are often encountered. Sociable coworkers can also increase anxiety if you're an introvert.

Engineer: proud term used by pretentious people. You will often encounter it in companies entrenched in a specific niche. The lack of competition usually leads to strange work ideals and preconceptions. The environment is usually filled with politics and managerial incompetence. And although most coworkers are introverts, everybody thinks very highly of themselves. So clashes are common, even for minor things. The pay is average. While the environment is great for learning and getting started, experienced people usually leave for greener and calmer pastures.

Of course, there are exceptions. But this is what I noticed so far based on jobs, interviews and people. Each of these titles has advantages and disadvantages. So I don't think one is better than the other. Picking one is a just a matter of what you currently want from your job.