Interactions between covariates Suppose I have a sample with two covariates whose levels are associated with some response; e.g., the mean height of a plant at 6 months is modeled by two factors:  cultivar of plant (say, $A$ versus $B$) and size of pot (e.g., "small" versus "large").  Furthermore, suppose that the truth is that cultivar $A$ in general grows taller than cultivar $B$ irrespective of the size of the pot in which it is planted.  However, the pot size has no effect on height at 6 months.
It so happens, though, that through random chance or some underlying hidden relationship, twice as many cultivar $A$ plants got the large pot as cultivar $B$, whereas in the small pot, the split is 50/50.  Now, if we naively do a $t$-test for $H_0 : \mu_L = \mu_S$ versus $H_a : \mu_L \ne \mu_S$, we would find significance not because the pot size has an effect, but because it happened that more $A$ cultivars were potted in the large pots.  Clearly such a test is not appropriate.
However, the question I'm faced with is this:  is it correct to say that cultivar and pot size have an interaction?  Note that pot size doesn't influence the outcome.  I know I should be using a 2x2 factorial design, but I have some difficulty interpreting the meaning of the tests:  If I compute a confidence interval based on $(\bar x_{AL} - \bar x_{AS}) - (\bar x_{BL} - \bar x_{BS})$, what is the related hypothesis, and is this test accounting for the fact that $A$ gets planted into large pots more frequently than $B$?
 A: The statistic 
$(\bar{X}_{AL} - \bar{X}_{AS})$ corresponds to an estimate of the effect of pot size when using cultivar A.  Similarly
$(\bar{X}_{BL} - \bar{X}_{BS})$ corresponds to an estimate of the effect of pot size when using cultivar B.  
Thus, the statistic
$(\bar{X}_{AL} - \bar{X}_{AS}) - (\bar{X}_{BL} - \bar{X}_{BS})$ 
corresponds to testing the hypothesis that the effect of large pots vs small pots when cultivar A is used is the same as the effect of large vs small when cultivar B is used.  This test statistic would be testing for an interaction.  It is looking to see if there is a different effect of large vs small depending on the type of cultivar, which is one way to interpret the definition of an interaction.
Notice that with a little algebra, the statistic can be rewritten as 
$(\bar{X}_{AL} -  \bar{X}_{BL}) - (\bar{X}_{AS} -\bar{X}_{BS})$, which is looking to see if the effect of cultivar A vs B depends on pot size.  These are two equivalent ways of how to think about what it means to test for an interaction between two variables.
You are correct that if you test the hypothesis $\mu_L=\mu_S$ that it will look like there is an effect of pot size, but the perceived effect is completely explained, as you point out, by the imbalance in the distribution of cultivar between the large and small pots.  The lurking variable, the variable that is actually driving the results, in this hypothesis test is cultivar.  It is not, however, a confounding variable.
A: 
However, the pot size has no effect on height at 6 months.

If there's an interaction, it's not correct to say 'pot size has no effect'; it's already been stated to have an interaction effect!

because it happened that more A cultivars were potted in the large pots. 

If pot size has no effect what difference does it make which pots the plants were in?
Your discussion seems to be self-contradictory. What do you mean to say?
