There are several ways to calculate and interpret the ICC for a mixed model. A useful thread is here. To calculate, it is the amount of variance from certain factor(s) divided by the total variance. That would be akin to your B and D calculations. This can be interpreted as the correlation of two randomly drawn units from the same grouping or as the amount of variance explained by those groupings, similar to an $R^2$.

This calculation (B): $$\frac{\sigma^{2}_{item}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation of scores for any item, regardless of the subject using the item.

Similarly, this calculation (D): $$\frac{\sigma^{2}_{subj}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation of scores from any subject, regardless of what item they are working on.

A combined calculation, such as this: $$\frac{\sigma^{2}_{subj}+\sigma^{2}_{item}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation between scores of the same person using the same item (in your data, this is 0.654).

Turning to the values you listed (0.488 and 0.165, respectively), a naive interpretation is the items you're using are modestly correlated regardless of who is using it, while the subject scores are mostly uncorrelated across all the items.

Finally, regarding your two additional calculations (A and C), I'm not aware of a useful interpretation of those values, however I have an open question on the topic.

There are several ways to calculate and interpret the ICC for a mixed model. A useful thread is here. To calculate, it is the amount of variance from certain factor(s) divided by the total variance. That would be akin to your B and D calculations. This can be interpreted as the correlation of two randomly drawn units from the same grouping or as the amount of variance explained by those groupings, similar to an $R^2$.

This calculation (B): $$\frac{\sigma^{2}_{item}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation of scores for any item, regardless of the subject using the item.

Similarly, this calculation (D): $$\frac{\sigma^{2}_{subj}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation of scores from any subject, regardless of what item they are working on.

A combined calculation, such as this: $$\frac{\sigma^{2}_{subj}+\sigma^{2}_{item}}{\sigma^{2}_{subj}+\sigma^{2}_{item}+\sigma^{2}_{res}}$$ could be interpreted as the correlation between scores of the same person using the same item (in your data, this is 0.654).

Turning to the values you listed (0.488 and 0.165, respectively), a naive interpretation is the items you're using are modestly correlated regardless of who is using it, while the subject scores are mostly uncorrelated across all the items.

Finally, regarding your two additional calculations (A and C), I'm not aware of a useful interpretation of those values, however I have an open question on the topic.