All vectors are coinitial?

Question

When representing vectors as directed lines with initial and terminal point:

Coinitial vectors- Two or more vectors that have same initial points are called coinitial vectors.

But aren't all vectors coinitial? Because vectors can be shifted (without changing orientation), so any vectors can be shifted to make their initial point same.

Isn't "coinitial vectors" a redundant unnecessary term?

Note: Coinitial vectors makes sense when vectors are represented in x-y or x-y-z vector plane/space. :Note

So is the definition of coinitial vectors this? -
Two or more vectors whose initial points are same in vector plane are called coinitial vectors.

But when not talking about vector plane (for example when illustrating vector addition, etc), aren't all vectors coinitial.
And what is right definition of coinitial vectors?

[Kindly edit the question if needed as I am new to maths, terms, definitions etc]

Answer 1

The vector field of airflow in a room is a collection of non-coinitial vectors. They cannot be coinitial because there is only one flow direction at any point in the room (excepting points where the flow is zero, since the direction is undefined at those points).

If you rigidly rotate a circle along itself, the displacement vectors are not coinitial because each point has only one destination, not more than one.

The velocity vectors of points on a sphere are not coinitial because all such vectors are parallel to the surface at their initial points. Different initial points give different tangent planes, so the initial point is crucial to determine the space of possible velocity vectors at a point on the sphere.

Answer 2

On manifolds, it is possible that two vectors are not coinitial.

The term "manifold" may sound scary; you are not even expected to know this when you first learn vectors. But we can look at some examples.

Think of a physical particle that is forced to move along a certain curve, maybe it is a bead moving through a curved wire. For example, the curve could be the unit circle $x^2+y^2=1$ in the plane. You can move the particle along the wire as you like. It doesn't have to be uniform circular motion. While the particle moves, you can talk about the velocity vector of the particle at some fixed time.

The velocity vector is required to have initial point being the position of the particle. If the particle is located at $(x,y)$, the velocity vector must have initial point $(x,y)$ itself. It may move at a certain velocity $\vec v$, or faster like $2\vec v$, or slower like $\frac{1}{3}\vec v$, or backward like $-\vec v$. These several vectors are all coinitial.

Meanwhile, if the particle is located at some other point $(x',y')$, you may not just move the vector $\vec v$ to the point $(x',y')$, and say the particle can move at velocity $\vec v$ at the point $(x',y')$.

For an explicit example, consider the two points $(1,0)$ and $(0,1)$. The possible velocities at $(1,0)$ go vertical, while the possible velocities at $(0,1)$ go horizontal. Velocity vectors at $(1,0)$ cannot be moved to $(0,1)$, and vice versa. So a vector at $(1,0)$ cannot be coinitial with a vector at $(0,1)$, when the particle is restricted to move along the circle.