Similarity and difference between $\color{red}{\text{least upper bound/greatest lower bound}}$, $\color{blue}{\text{maximal/minimal}}$, $\color{green}{\text{maximum/minimum}}$.
Given a partially ordered set $\left(P,\le\right)$ and a subset $S$ of $P$:
An element $ g \in P$ is the $\color{red}{\text{least upper bound/greatest lower bound}}$ of $S$ if :$$\forall s \in S : s\le g \;\;\text{and for every upper bound u of S} : \;\;\;g\le u \;\;\;\ $$
$$(\forall s \in S : g\le s \;\;\text{and for every lower bound u of S}: \;\;\;u\le g )$$
An element $ g \in S$ is the $\color{blue}{\text{maximal/minimal}}$ of $S$ if :$$\forall s \in S : s\le g \;\;\;\ (s\ge g)$$
Equivalently if there does not exist any $s\in S$ such that:
$$g\le s\ \text{and}\ g\ne s \;\;\;\ (g\ge s\ \text{and}\ g\ne s)$$
An element $ g \in S$ is the $\color{green}{\text{maximum/minimum}}$ of $S$ if :$$\forall s \in S : s\le g \;\;\;\ (s\ge g)$$
The similarity between all of them is that it may happen that neither least upper bound/greatest lower bound nor maximal/minimal nor maximum/minimum exist.
The similarity between maximum/minimum and maximal/minimal is that if they exist then they both belong to the set $S$, although a least upper bound/greatest lower bound may not belong to the set $S$ and their difference is that it may happen to have more than one maximal/minimal but maximum/minimum or least upper bound/greatest lower bound is always unique.
If a set has a $\color{green}{\text{maximum/minimum}}$ then its $\color{blue}{\text{maximal/minimal}}$ are unique and every maximum/minimum is a maximal/minimal but not vice versa, also every maximum/minimum is a least upper bound/greatest lower bound but not vice versa. it can be concluded that a maximum/minimum are a stronger form of maximal/minimal and the condition incomparability never can happen.
are my conclusions right?
First thing, some of your definitions seem off to me, for instance I do not see any difference between the definitions you gave of "minimal/maximal" and "minimum/maximum" (and also, the property stated after "equivalently" for minimal/maximal also seems off). I interpret the order relation $\leq$ as meaning "lower or equal". So let me first go through the definitions:
Any of these notions isn't guaranteed to exist. For instance, the natural integers $\mathbb N$ with the natural order, seen as a subset of itself, is a (totally) ordered set without any upper bound (but with a minimum, that is $1$, or $0$ depending on the conventions). The interval $(0,1)$ inside $\mathbb R$ is a subset with both least upper bound, greatest lower bound, but no maximal nor minimal element. The subset $\{2,4,6\}$ of $\mathbb N$ ordered by divisibility (that is, $x\leq y$ if and only if $x$ divides $y$) has to maximal elements $4$ and $6$, but no maximum as $4$ and $6$ are not comparable. It also has a minimum, that is $2$.
As you said, least upper bounds, greatest lower bounds aren't required to be part of the subset $S$, if they exist (but they may be). Maximal/minimal, maximum/minimum elements must be part of the subset $S$. We may have many maximal/minimal elements, but maximum/minimum and least upper bounds, greatest lower bounds are all unique when they exist (that is because they are comparable to every element of $S$, check it !).
If a set has a maximum/minimum, then it is its unique maximal/minimal element. The converse is false : a set can have a unique maximal element that is not the maximum. For instance, consider $\mathbb N$ which has no maximal element, and add an element $e$ that is not comparable to any of the integers. Then $e$ is trivially a maximal (and minimal) element of $\mathbb N \cup \{e\}$, it is the only maximal element, but it is not the maximum (the maximum doesn't exist).
If a subset has a maximum, then this maximum is the least upper bound, and similarly for minimum with respect to greatest lower bound. But again, the converse is easily seen to be false.
In some sort of conclusion, maximum/minimum is the strongest notion to work with. Least upper bound/greatest lower bound and maximal/minimal can be seen, if you want, as two different ways to "generalize" the definition of maximum/minimum. It is weaker, but it is more likely to exist in general.