Under the definitions you've listed, "association" and "relationship" would not be interchangeable. However, I would argue that a better use of the term "relationship" would make them fairly synonymous in this application. I think that your teacher was making an important, and correct, point about correlation and regression, but that the way it was done (at least according to your memory) used the term "relationship" in a non-standard way. I think you are on solid footing to make the claim as you do in your last paragraph. For more info on the asymmetrical vs. symmetrical nature of regression and correlation, see here.
No; correlation is not equivalent to association. However, the meaning of correlation is dependent upon context.
The classical statistics definition is, to quote from Kotz and Johnson's Encyclopedia of Statistical Sciences "a measure of the strength of of the linear relationship between two random variables". In mathematical statistics "correlation" seems to generally have this interpretation.
In applied areas where data is commonly ordinal rather than numeric (e.g., psychometrics and market research) this definition is not so helpful as the concept of linearity assumes data that has interval-scale properties. Consequently, in these fields correlation is instead interpreted as indicating a monotonically increasing or decreasing bivariate pattern or, a correlation of the ranks. A number of non-parametric correlation statistics have been developed specifically for this (e.g., Spearman's correlation and Kendall's tau-b). These are sometimes referred to as "non-linear correlations" because they are correlation statistics that do not assume linearity.
Amongst non-statisticians correlation often means association (sometimes with and sometimes without a causal connotation). Irrespective of the etymology of correlation, the reality is that amongst non-statisticians it has this broader meaning and no amount of chastising them for inappropriate usage is likely to change this. I have done a "google" and it seems that some of the uses of non-linear correlation seem to be of this kind (in particular, it seems that some people use the term to denote a smoothish non-linear relationship between numeric variables).
The context-dependent nature of the term "non-linear correlation" perhaps means it is ambiguous and should not be used. As regards "correlation", you need to work out the context of the person using the term in order to know what they mean.
Best Answer
They may sometimes be used as if they mean the same thing but correlation is more specific, and association is more general, with relationship being between the two.
Correlation means that they move together (positive correlation indicates increasing and decreasing together, negative correlation means they move in opposite direction). Linear correlation is more specific still; then they move in proportion, not just in the same (or opposite) direction.
A relationship suggests that as one variable changes the other tends to change as well. For example, two variables may have a quadratic relationship. This may occur with correlation or variables may be related but uncorrelated. That is, unless accompanied by some qualifying term, I tend to interpret "relationship" to essentially imply "functional relationship". However the word "relation" in the mathematical sense is more general and many people often use it more broadly without qualifying it. For example, imagine points scattered about a set of concentric ring-shapes. I probably wouldn't say that the variables were related (at least not without adding something to say that they weren't functionally related) but some people would happily do so.
Association may be more general still; (to me at least) it suggests almost any form of dependence between the variables. For example, imagine two variables where all the points tend to lay within small discs of probability (and not outside them), but where these discs are scattered about in a way that looks random. Then I would usually avoid the term "relationship" (qualified or not) and stick to associated or dependent.
Further example: If you look at this image (from the Wikipedia article on Correlation and dependence; the first row displays different amounts of linear correlation, while the last row shows variables that are uncorrelated but dependent. With the the first and fourth, I'd definitely say that the variable on the y-axis had a relationship with the one on the x-axis and I'd call all of them associated. (I might use "related" for the 5th and 6th but I'd tend to want to clarify that they weren't functionally related.)
However, as with many terms in statistics, there is variation in how people use terms like relationship and association; you may see them used almost interchangeably, with association used less generally than just any form of dependence and/or relationship used more generally than I have.
If a variable is correlated with another, then it could also be called related or associated with it. However, a variable may be related to another, or associated with another, but not be correlated to it.