When referring to the absolute value I have said Modulus. People then correct me and say that that is the operation where you find the remainder. Then I get confused because I've seen people say modulo for the remainder operation. Also if my first statement is correct why do we have two words for absolute value?
Difference Between Modulus, Absolute Value, and Modulo
terminology
Related Solutions
This is a tremendously common confusion to have, and in my experience, people are notoriously bad at explaining this concept. I'm sorry that you had to deal with people who were abrasive in addition to poor expositors.
In an arbitrary vector space, you cannot talk about components. They actually don't exist. Now, you can impose them on a finite-dimensional space by providing a bijective linear transformation from the arbitrary vector space to $F^n$, but then they're just that: an imposition, because any other bijective linear transformation will choose different would-be "components".
Components exist in $F^n$ because of the actual nature of the objects involved. So you don't need a basis, you can just look at an arbitrary object $(a,b,\dots,n)$, and find any of its components, because they're built into the object. This can be confusing because we also write coordinate vectors in this way, and when the basis is the standard basis, there is no difference between the components and the coordinates. However, in any other basis, there will be a difference.
(Edit: Val made an important point in the comments. I should have been more careful when I said there was "no difference". The fact is that coordinates and components are never conceptually the same, but I meant to say that in the standard basis case they will be numerically equal.)
Lacking a basis at all, you might want to say that $F^n$ still has coordinates implied by its components. But, in my opinion, this seems silly, since you cannot do the same in other spaces.
So the short answer is: Yes, there is a difference, because components are part of the objects.
As for your "collection of vectors" notion, they are basically the same. But it is easy to imagine a collection of vectors which is not a vector space: for example the circle in $\mathbb{R}^2$. This is definitely a collection, and the objects in it are definitely vectors, but it is not a vector space.
What I assume you meant by "collection" was what we might call a "meaningfully structured collection", and the meaningful structure is described precisely as an abelian group over which elements can be scaled by objects in a field. In that sense, your notion is correct, though a bit less transparent.
A constant is something like a "number". It doesn't change as variables change. For example $3$ is a constant as is $\pi$.
A parameter is a constant that defines a class of equations. $$\left(\frac xa\right)^2 + \left(\frac yb\right)^2 = 1$$ is the general equation for an ellipse. $a$ and $b$ are constants in this equation, but if we want to talk about the entire class of ellipses then they are also parameters -- because even though they are constant for any particular ellipse, they can take any positive real values.
A variable is an element of the domain or codomain of a relation. Remember that functions are just relations so the input and output of functions are variables. For example, if we talk about the function $x \mapsto ax +3$, then $x$ is a variable and $a$ is a parameter -- and thus a constant. $3$ is also a constant but it is not a parameter.
A "known" variable is typically a value that the conditions of the problem dictate the variable must take. For example if we are discussing an object an free fall, then acceleration is a variable. But physics puts a constraint on the value that that variable may take -- acceleration in free fall is $a=g\approx 9.8$. Thus, though $a$ may be defined as the input of a function, it must take a "known" value. Thus it is a known variable.
The Pythagorean theorem states that $a^2 + b^2 = c^2$ for sides $a,b$ and hypotenuse $c$ of a right triangle. These are parameters -- thus they are also constants.
Best Answer
They mean differently.
$\color{green}{\Large\bullet}$ Absolute value of $x = |x|$ and is equal to $x$ if $x \geq 0$ or is equal to $-x$ if $x < 0$.
$\color{green}{\Large\bullet}$ Modulo, usually refers to the type of arithmetic called modulo arithmetic. For example, because $13 = 4\times 3 + 1$, we write $13\ \equiv\ 1\ (\textrm{mod}\ 3)$. In common mathematical language, it is taken as "$13$ is congruent to $1$ modulo $3$".
$\color{green}{\Large\bullet}$ Modulus refers to the magnitude/length of a vector.
Added
How about “An introduction to the theory of Numbers – by Niven Zuckerman” and “Pure Mathematics I & II by F. Gerrish”?
Those names in question have been commonly used by others and sometimes even interchangeably. But, in the books mentioned above, they are clearly and distinctly defined.
The only confusion comes from the “$|…|$” sign, which has been used both for the absolute value of a number and also as the modulus of a vector. Therefore, some used the “$|| … ||$” for the latter to make the meaning distinct. Some don’t even bother when the context is clear or when the readers should be able to distinguish their difference.