Four vectors and coordinates Hartle, gravity. Chapter 5

A four-vector is defined as a directed line segment in four-dimensional flat spacetime in the same way as a three-dimensional vector (to be called a three-vector in this chapter) can be defined as a directed line segment in three-dimensional Euclidean Space.

For vectors in 3D, we know that vectors exist independently of the coordinate system used to measure them. Like a position vector. It exists there, it's the tip at a point in the 3D space.
We may choose different coordinates to write the vector but all of them say the same thing. They all are representing a vector whose tip ends at the same point.
So similarly should I as the book suggests, think of spacetime as being out there independently of the coordinate system, and for example, a displacement vector in spacetime is just the line segment from one point to another?
Now once I choose a coordinate system I get components of the vector  ( which is independent of the coordinate system)  in that coordinate system.
If I apply a Lorentz boost, I'm actually choosing another coordinate system which thus changes the coordinates of the vector. However, they actually represent the same vector whose tip ends at the same point, although that point has different coordinates in different frames.
Is that a correct understanding?
 A: Yes, this is exactly the way I explain four vectors to students new to special relativity, and it neatly explains the invariance of the proper time/proper length. Since it's the same vector, regardless what coordinate system we use to describe it, the length of the vector has to be the same in all coordinate systems. So when we define the length as:
$$ ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 $$
The proper length $ds$ is an invariant. Then we derive the Lorentz transformations as (one of the) transformations that preserve the proper length.
A: Very much so, we have an abstract tensor $\mathbf{T}$ which "lives" on the manifold's tangent space and if we choose a coordinate system with the basis $$\{\partial_\mu\}, \ \  \mu = 0,1,2,3,$$ we can write out the abstract tensor as
$$
\mathbf{T} = T^\mu \partial_\mu
$$
where we denoted $T^\mu$ as the components of this record, we can also choose another coordinate system $\{\partial^\prime_\mu\}$ in which we could also write out the abstract tensor $\mathbf{T}$ in the same sense. These two coordinate systems (since they are arbitrary) can surely be choose such that one is transformed into the other with the Lorentz transform.
So to answer you question: yes you are correct.
