The SE1 can indeed be used as a substitute (in the US) for the PE exam and you can then be licensed as a PE. Technically, you cannot call yourself an SE if you only pass the SE1 exam. Most states in the US only acknowledge the PE designation OFFICIALLY and usually allow the individuals to refer to themselves as a "structural" engineer, "professional" engineer, etc. as indicative of their own designated area of expertise.
In fact, most states leave it up to the individual to ethically practice only within their designated area of expertise. If I were to attempt to design a sanitary sewer, being a structural engineer, it would be a violation of the state licensing law in that I was practicing out of my area. I am responsible for keeping myself involved only in areas where I am competent.
Once you pass SE2, you then, technically can call yourself an SE, even though most states do not legally distinguish between SE and PE in terms of titles. Out in the western states, with higher seismic zones, there is a legal distinction in that an SE has indeed passed the SE2 (and in some cases the SE3 or western states exam) and holds the legal title of SE (i.e. California).
Most all states allow PE's to design structures if that's their area of expertise - whether they passed the PE or SE1 exam.
Illinois does NOT allow PE's to design structures. You must pass the SE2 and be licensed as an SE. California allows PE's to do structural up to a point - High rise, schools and hospitals are for SE's only. Washington state allows the LOCAL governing entity (the city) to designate when an SE is required over a PE.
There's probably a lot of other laws in other states - each is different.
You cannot pass the PE and then skip SE1 to take SE2.