Ecchi - Teenage near sex fantasy.
Adult - The Adult world in context
Mature - Psychological Maturity in humans
Smut - Sex in a nutshell
Basically this, yeah. Basically, how you think about it, "smut" is the only title that has to contain sex in order to not qualify as false advertisement. Ecchi is a tag where they talk about and hint about sex a lot, but it never actually happens. (At least not on screen.) Those are the two boarders. Smut always contains sex, Ecchi never does.
The two in the middle, "adult" and "mature" are both headings that are perfectly compatible with sexual content, but don't need sexual content. "Adult" means it's intended for an adult audience. This can mean sex, but it can also mean that it just talks about things that would interest adults a lot more than it would younger audiences. Such as, maybe a very realistic portrayal of what it's like to raise a child. This would qualify as "adult," but most likely would not have all that much sex in the story.
"Mature" means that it entertains your brain by challenging it with various concepts that are hard to tackle and require a mature mind-set. For instance, letting the scum-bag go without a beating because you realize that while giving that scum-bag the smack-down now would give a bit of momentary catharsis, it would actually not be good for you in the long run. I actually did a bit in my story where my MC made sure to yell down her best friend who was trying to protect her and in the meantime heal the guys who might be trying to kill her. She did this because her life literally depended on whether or not she could generate good optics in front of the members of the village. And, while they were not there when she was doing this, she knew stories would spread whichever choice she made.