I think the elements of your suggested definition are ok, but I would want to split them up and put a lot more explanation in between.
I would perhaps start by saying that there are different ways of thinking about vectors, and that while they are all the same in some sense (and so it's reasonable to use the same word), it might not be obvious to everyone until they get used to switching the way they think about them.
Then, according to time/syllabus, I'd introduce the different ways of looking at vectors in the contexts that each is used: position vectors in coordinate geometry, free-floating arrows in coordinate-free geometry, direction/magnitude in mechanics, $n$-tuple of objects in sequences or programming.
For each one I'd get them doing questions in that context. After that, I'd move on to questions that involve changing between two of the descriptions, eg Cartesian and polar coordinates. Doing such questions should allow students to mentally connect the definitions, although some won't manage that level of understanding.
The reason I think your definition is potentially confusing is that it mixes the different viewpoints, at a point when the students don't have the understanding to see the connections. If you're just working with $n$-tuples, direction and length don't have any meaning, particularly if the entries might not be real numbers, and making the shift in thinking is likely to take longer than reading one sentence. Using Pythagoras only makes sense when you have some kind of coordinates, and seeing why it has anything to do with something other than triangles will take time too. The students need to get their heads around the different forms before they can put the different sentences together, if they are to be happy with a definition presented that way.