Dimension in mathematics is defined as the number of independent components of a structure (with some generalizations, like fractal dimensions). Now which number of dimensions you assign to e.g a discrete image (as in picture) depends on what aspect of the image you are describing. As a vector space where each pixel is a component, you get as many dimensions as pixels. The same for a discrete signal, the number of samples is your dimension of the vector space.
However, sometimes you're rather interested in the manifold the object is a function on. The real line is a 1-dimensional real manifold, and a real function on it would be a signal. So you can call the signal "one dimensional", explicitly referring to this property. Sometimes your function may map to the complex numbers or to R^2 (still from the real line!) instead, and you might want to call that a "two dimensional signal", and that's also correct. However, a two dimensional signal may also just be a function on a two dimensional manifold, like a plane.
So there's a lot ambiguity in verbal descriptions, which is why to be clear you should use exact mathematical notation. Opposite to common belief it was not just invented to torture students.
