You might want to look at how vectors are introduced to your students in previous coursework. At my school (a community college in California), it would be something like this:
The remedial trig course includes vectors in the course outline, but not all students take that course, and if they do, the instructor will often skip the topic.
A first semester physics course spends a lot of time on vectors. Probably some but not all of your students have taken such a course.
The sophomore multivariate calculus course introduces vectors from scratch on the assumption that students have never seen them before.
The linear algebra course spends most of the semester teaching students algorithms for manipulating matrices and row and column vectors. I doubt that students connect these manipulations in any way to any possible geometric interpretation.
If you ask your students to define what a vector is, you will probably get a variety of answers, depending on which of these experiences they've had. My guess it that some will say it's a thing that has both a magnitude and a direction (a common definition in physics textbooks), while others will say that it's an n-tuple of numbers (basically the only representation they ever seem to work with in the math courses). Neither of these is quite the same as the definition that would apply best to your example, which is that a vector is an equivalence class of displacements between points in space.
When I teach my students about vectors in freshman physics, I find that they need a lot of encouragement to think about vectors as "portable." For example, they may draw a free-body diagram in which two force vectors act on an object, and the force vectors are drawn tail to tail. Then if they want to sketch graphical addition of the vectors, they have to slide them around into tip-to-tail position. This is not natural to them, nor is it natural to them to understand that the vectors have to be slid around without rotation (parallel transported, in fancy terminology).
In my office hours, I spend a lot of time demonstrating this with pens laid on the tabletop. I place two pens on the tabletop, tail to tail, and ask the student if they can slide them around so they're tip to tail. Usually they don't understand how to do this. Then I usually take one pen, move it around some path and bring it back to the start again, to illustrate the idea of sliding without twisting. I may also explicitly demonstrate twisting, in order to show that it's not allowed.
In general, students usually do not get much practice with the conceptual aspects of vectors. For example, I have them do this think-pair-share exercise:
In example 4 [computing the components of a vector given in magnitude-direction form], we dealt with components that were negative. Does it make sense to classify vectors as positive and negative?
This may seem extremely elementary to a mathematician, but for many students it's very difficult.
For the particular example you give, of converting between points and vectors, note that for students who have learned about vectors in a physics class, their stock of mental examples of vectors probably has a couple of primary examples in it: force vectors and displacement vectors. For force vectors, it isn't even true that you can convert back and forth between a point form and a vector form, if the point is to be a point in the ordinary Euclidean space that we swim around in. It could of course be a point in a more abstract "space" of force vectors, but this is not a level of abstraction that would occur to students at this level.