As a software engineer, do you really need to be a seasoned programmer?
The title Software Engineer is used in many places to mean "programmer." There's more to software engineering than just programming: software engineering is about the process of creating software, and that includes programming but isn't limited to just programming. Software engineering considers the processes you create to develop requirements, detect and eliminate defects, produce updates, etc.
Is it true that software engineers don't code but just make diagrams,
functional specs. and othe related documents?
No. At least, I've never met a software engineer who didn't write code. What you're describing sounds more like the role of a business analyst, whose job it is to look at a problem from a business perspective and figure out how to build a system that (both hardware and software) that addresses the businesses needs.
I also noticed that there are no SE standards or board exams to pass
and its really a dynamic and situational job.
That's true in many places, but it may be changing. See: NCEES introduces PE exam for software engineering. According to Wikipedia, if you want to call yourself a software engineer in some places you need a license or certification.
So basically I can just proclaim myself as a software engineer based
on experience and product?
Yes, but you can also proclaim yourself King of the World if you want. At some point, you have to back it up or people will eventually get the idea that you don't know what you're talking about.