It can be futile to fight against shifts in language. But
parameter does not mean variable
In classical statistics, which in this case starts precisely with R.A. Fisher who first used the term with this meaning, a parameter is an unknown constant to be estimated, say a population mean or correlation. In mathematics, there are related but not identical meanings, as when a curve is given parametrically. In many sciences, parameter is just another word for a measure (itself a term dense with mathematical meaning), property or variable, say length or conductivity or porosity or virtue, as the case may be. Naturally, an individual's length or virtue is unknown before it is measured. but statistically minded people can be bemused by its use for a set of such measurements. In ordinary or vulgar parlance, parameters (almost always plural) often mean the limits of something, say a personal relationship or a political policy, perhaps stemming from some original confusion with perimeter. With high prior probability it is to be presumed that Bayesians will speak for themselves on their own usages (grateful nod to @conjugateprior).
skewed does not mean biased
For a century or more, skewness has had a specific statistical sense of referring to asymmetry of distributions, whether assessed graphically, measured numerically, or presumed theoretically as a matter of faith or hope. For much longer, or so it may be guessed, bias has meant being wrong on average, which -- so long as we know the truth, meaning a true or correct value -- can be quantified as systematic error. Skewed in ordinary language has a common sense of being warped or distorted, and thus of being incorrect, wrong and so also biased too. That sense (so far as I have noticed, only quite recently) has begun filtering back into statistical discussions, so that the original meaning of skewness is in some danger of being blurred or submerged.
correlation does not mean agreement
Correlation has attracted several precise senses in statistics, which have in common an idea of a bivariate relationship perfect in some precise sense: the leading cases are linear and monotone relationship. It is often diluted, even in statistical discussions, to mean almost any kind of relationship or association. What correlation does not mean, necessarily, is agreement: thus $y = a + bx$ implies Pearson correlation of $1$ or $-1$ so long as $b \ne 0$, but agreement $y = x$ requires the very strict condition $a =0, b= 1$.
unique does not mean distinct
It is quite common to talk about the distinct values of data as unique, but unique is still ideally better preserved as meaning occurring once only. My own guess is that some of the blame stems from the Unix [sic] utility uniq and its imitators, which reduce possibly repeated values to a set in which each value really is unique. The usage, on this guess, conflates input and output of a program. (Conversely, if we talk of duplicates in data, we rarely restrict ourselves to doubletons that occur precisely twice. The term replicates would make more sense linguistically but has been
pre-empted for deliberate replication of controls in experiments; the resulting response values are usually not at all identical, which is much of the point.)
samples are rarely repeated
In statistics, a sample includes several values, and repeated sampling is a high theoretical virtue, but one rarely practised, except by simulation, which is our customary term for any kind of faking in silico. In many sciences, a sample is a single object, consisting of a lump, chunk or dollop of water, soil, sediment, rock, blood, tissue, or other substances varying from attractive through benign to disgusting; far from being exceptional, taking many samples may be essential for any serious analysis. Here every field's terminology makes perfect sense to its people, but translation is sometimes needed.
error does not usually mean mistake; as Harold Jeffreys pointed out, the primary sense is erratic, not erroneous.
Nevertheless, we should be wary of our own sins or quirks of terminology:
expected values or expectations (for means over the possible outcomes) may not be what you expect at all, and could even be impossible: in tossing a die fairly with outcomes 1 to 6, the expected value is 3.5
regression is not going backwards
stationary does not mean immobile or fixed
confidence has nothing to do with anyone's mental or psychological state
significance has only sometimes its everyday meaning
exact is often an honorific term, referring to a conveniently tractable solution or calculation rather than one appropriate to the problem
right-skewed distributions to many look skewed left, and vice versa (and the terminology of right and left for skewness assumes that you are looking at something like a conventional histogram, with horizontal magnitude axis)
the lognormal is so called because it's an exponentiated normal
but the lognormal is more normal than the normal
the Gaussian was discovered by De Moivre
Poisson didn't discover the Poisson, let alone Poisson regression
the bootstrap won't help you with your footwear
the jackknife doesn't cut
kurtosis is not a medical condition
stem-and-leaf plots don't refer to plants
a dummy variable is useful, not pointless or stupid
who on Earth (or anywhere else) thinks that heteroscedasticity is really a preferable term over unequal variability?
robust now has at least two major technical meanings for different groups, neither of which inhibits its frequent use, even in technical discussions, to mean merely something like "asserted to behave well"
IV now has at least two major meanings for different groups
factor now has at least two major meanings for different groups
normalize and standardize have uncountably many meanings (we really need to standardize there)
versus describing a graph means vertical variable versus horizontal variable, unless it means the opposite
and (last but not least, to coin a phrase) statistics has at least three major meanings.
Notes:
Despite any appearances to the contrary, I think this is a good, serious question.
Fashions shift. Well into the twentieth century, it seems that many people (no names, no pack-drill, but Karl Pearson could be mentioned) could only invent terms by reaching for their Greek and Latin dictionaries. (It would be unfair not to give him credit for scatter plot.) But R.A. Fisher did hijack many pre-existing English words, including variance, sufficiency, efficiency and likelihood. More recently, J.W. Tukey was a master in using homely terms, but few should feel distress that sploms and badmandments did not catch on.
One comment is based on recollection of "Life is [...] Multiplicative rather than additive: the log normal distribution is more normal than the normal." Anon. 1962. Bloggins's working rules. In Good, I.J. (Ed.) The scientist speculates: an anthology of partly-baked ideas. London: Heinemann, 212-213 (quotation on p.213).
Best Answer
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.