Alright, I'll add what I've come to understand it to be.... though I know I have a pretty uninformed opinion. I have a better understanding of what it means in the DR. I’m sure others are experts in what is means in other countries.
I think your point 5, is the first meaning of the word mambo, and the reason the word gets re-used so many times in so many different genres.
In bachata it’s the high-energy jam/improv/instrumental solo section of a song. You can usually distinguish it by the guira playing the ‘caballito’ (galloping) rhythm. You don’t often hear it much on studio recorded versions, since it is understood to be a live thing, though some artists like Anthony Santos do try and use a bit of it in their recordings.
In merengue your point 3 Covers most of it, but eventually it specifically came to refer to the driving beat that tambora plays during that section. It’s a simpler rhythm than most other merengue rhythms because of the high-energy, the speed and the improv solos. After that entire merengue songs were written using that rhythm throughout the whole song, rather than for just a section, and the genre of merengue mambo was born (point 4). Had it’s hey-day from about 2004-2010, whereafter merengue electronico came around and brought the speeds and energy right back down.
I’ve seen & heard the Cuban version (point 1 and 2), and naturally sneak in the odd mambo move when dancing casino, or when it comes up in rueda. Don’t have the depth of knowledge to explain it, but I mean something like this
In the States is where I get most confused, I know it’s a thing, but don’t know enough to know where one genre starts and another ends.
It’s just helps to be aware that it means different things in different places. I’ve seen someone ask a Cuban DJ for a mambo, thinking they would be getting something really New York sounding, and then getting annoyed that they were playing this weird old music. I’ve also seen a Dominican asking a Colombian DJ for a mambo and then complaining that the DJ didn’t play a single merengue all night.