Nominal vs Interval
The most classic "correlation" measure between a nominal and an interval ("numeric") variable is Eta, also called correlation ratio, and equal to the root R-square of the one-way ANOVA (with p-value = that of the ANOVA). Eta can be seen as a symmetric association measure, like correlation, because Eta of ANOVA (with the nominal as independent, numeric as dependent) is equal to Pillai's trace of multivariate regression (with the numeric as independent, set of dummy variables corresponding to the nominal as dependent).
A more subtle measure is intraclass correlation coefficient (ICC). Whereas Eta grasps only the difference between groups (defined by the nominal variable) in respect to the numeric variable, ICC simultaneously also measures the coordination or agreemant between numeric values inside groups; in other words, ICC (particularly the original unbiased "pairing" ICC version) stays on the level of values while Eta operates on the level of statistics (group means vs group variances).
Nominal vs Ordinal
The question about "correlation" measure between a nominal and an ordinal variable is less apparent. The reason of the difficulty is that ordinal scale is, by its nature, more "mystic" or "twisted" than interval or nominal scales. No wonder that statistical analyses specially for ordinal data are relatively poorly formulated so far.
One way might be to convert your ordinal data into ranks and then compute Eta as if the ranks were interval data. The p-value of such Eta = that of Kruskal-Wallis analysis. This approach seems warranted due to the same reasoning as why Spearman rho is used to correlate two ordinal variables. That logic is "when you don't know the interval widths on the scale, cut the Gordian knot by linearizing any possible monotonicity: go rank the data".
Another approach (possibly more rigorous and flexible) would be to use ordinal logistic regression with the ordinal variable as the DV and the nominal one as the IV. The square root of Nagelkerke’s pseudo R-square (with the regression's p-value) is another correlation measure for you. Note that you can experiment with various link functions in ordinal regression. This association is, however, not symmetric: the nominal is assumed independent.
Yet another approach might be to find such a monotonic transformation of ordinal data into interval - instead of ranking of the penultimate paragraph - that would maximize R (i.e. Eta) for you. This is categorical regression (= linear regression with optimal scaling).
Still another approach is to perform classification tree, such as CHAID, with the ordinal variable as predictor. This procedure will bin together (hence it is the approach opposite to the previous one) adjacent ordered categories which do not distinguish among categories of the nominal predictand. Then you could rely on Chi-square-based association measures (such as Cramer's V) as if you correlate nominal vs nominal variables.
And @Michael in his comment suggests yet one more way - a special coefficient called Freeman's Theta.
So, we have arrived so far at these opportunities: (1) Rank, then compute Eta; (2) Use ordinal regression; (3) Use categorical regression ("optimally" transforming ordinal variable into interval); (4) Use classification tree ("optimally" reducing the number of ordered categories); (5) Use Freeman's Theta.
Best Answer
You could use Spearman's, which is based on ranks and therefore OK for ordinal data. You would then have six results.
If you want to take a different approach, you could get complex and look at a multilevel model, with subject being repeated. It sounds like "accuracy" would depend on "preference". So, a mixed model could look at that and account for the non-independence of the data. But, as noted, that's a much more complex model to implement.