It's 1:1 with the relations in the equations up until the analogy of warped Euclidean space changing the vector at which point the description is functionally very similar but relativity follows very different (but also somewhat similar in a way) mathematical mechanics to the vector changing.

The "spacetime speed vector" is more formally the four-velocity and it's true that the norm of this 3 component space 1 component time vector is strictly tied to c. At the same time the four-velocity doesn't actually mathematically behave like a euclidean vector space vector where you can just add another like vector describing the effects of the warping and call it a day. In reality you have to run it through the metric tensor first (some function for the given instance that describes the geometry of warped spacetime) to get things in a coordinate space that is usable. Once you have that you actually have to run it through the geodesic equation to see what the acceleration will be as using the mapped four-vector alone will only tell you about the current velocity components in your coordinate space not the effect of the spacetime warping on something in them. These kinds of differences are the bits I swept under the rug as "nuance in how things actually change" but the net concept of the four-vector shifting components due to the warping of spacetime as an object moves along its world line is 100% the net result.

Also I can't really take credit for the method of explanation, just some of the simplified wording. I do find this explanation not only infinitely more accurate but actually easier to understand than the damn rubber sheet analogies or even improved/3D space warping analogies as they still leave out the time portion of the spacetime gradient which actually plays a bigger role in these examples.