"Dimension" often means the minimum number of coordinates needed to specify a point in coordinate space. In this context, a "coordinate" is a number or name referring to an axis. So we might say that $x$, $y$, and $z$ are coordinates, while $3$ is the coordinate system's dimension. But these terms are often overloaded.
For example, people often say things such as the "$x$ dimension". Here "dimension" is the same as "coordinate": it is an axis rather than the minimum number of coordinates. And people often refer to objects such as $(1, -10, 7)$ as "coordinates". Here, a "coordinate" is no longer an axis but a tuple. Furthermore, people might even refer to a single value in a tuple as a "coordinate", e.g. "In $(10, 5)$, the $x$-coordinate is $10$".
So my questions are:
- Is a "dimension" a number or an axis or both depending on context?
- Is a "coordinate" an axis, a tuple, or a single value from the tuple?
If you want each word to have just one meaning, use dimension only for the cardinality of a basis of the vector space, i.e. a number of Cartesian coordinates needed to specify a point, and coordinate for one of the numbers used in such a specification. An axis is then a locus of all but one of the coordinates being zero, e.g. $y=z=0$ is the $x$-axis in one parametrization of $3$-dimensional space. All the numbers as a tuple would be the point's coordinates. A "coordinate" definitely shouldn't mean an axis.
However, there is one downside to the above approach. How do you refer to a locus of the form $y=a,\,z=b$ for $a,\,b$ not both $0$? I don't think anyone has a name for these; we just say such lines are parallel to the $x$-axis. When people talk of the $x$-dimension, they probably want to compare points differing only in $x$ (i.e. lying on such a line), or to talk about what happens as you move along such a line. It's probably best to talk just about varying $x$.