I'm sorry but anyone that is making a distinction in skill level or seniority between these titles is wrong. Perhaps this is a cultural bias on their part, but IMO it's important to take words at face value. 'Web Development' is a subset of 'Software Development,' just as 'Mobile App Development' or 'Embedded Software Development' are subsets of it as well. A software's platform should not be the sole determiner of how complex or difficult the problems being solved on it are. If it were, though, the web is one of the more complex and difficult mediums to develop for as the developer has far less control over the device and manner of consumption than other environments.
Now bring PHP, Ruby, Python, Java, etc into the mix. PHP gets picked on because it has a very low barrier to entry / learning curve, but that doesn't mean there aren't world-class frameworks and whole armies of *very* experienced and skillful senior software engineers that use it. PHP was made for the web and so may be more associated with 'web development,' but the others are also used in that context. And when they are used in that context, they are not automatically simpler or more fit for more junior developers. That's just nonsense.
I introduce myself as a 'Web Developer.' I love the web and it accurately describes my focus under the umbrella of software development, My experience runs the gamut between frontend development to backend API development to web application devops. Over the course of my career I've had titles all over the map: UI Developer, Frontend Developer, Web Developer, Software Engineer, Senior Software Engineer, and even just Lead Developer, etc. The title is meaningless; what matters is what you do.
Some people get caught up in thinking that 'Software Engineer' sounds more legit than other titles for some reason. I get that the word 'engineer' sounds impressive, but it's all smoke and mirrors and should be seen as such. The truth is that title is extremely broad and encompasses a plethora of specializations. It's like being more impressed with someone being called an 'Attorney' than being specifically called a 'Divorce Lawyer'. Both are lawyers; the former may sound more formal and encompasses many specialties, but a 'Divorce Lawyer' is no less a lawyer than any other attorney.
So to answer your question: I prefer whichever title suits me for a given purpose. If someone asked all software engineers to raise their hand, I would undoubtedly raise my hand. But since everyone is familiar with the web, I find it easy to describe the work I do as web development to people I meet because it conveys my specialty within the realm of software development; I'm the guy that makes the websites you use work.