Difference of sklearns accuracy_score() to the commonly accepted Accuracy metric
Your "commonly agreed on" and "standard" accuracy is meant for binary classification, in which case it agrees with the other formula from sklearn.  In that case, "positive/negative" refer to the two classes, so this is also a little different from your version.

Your version of it is a sort of average of (the "mediant" of) the implicit one-vs-rest classifiers.  As such, your score is meaningful, but will generally be larger than the actual common multiclass accuracy metric.  For a balanced problem, a constant classifier will get a mediant-of-OVR-accuracy score of $(n-1)^2/n^2$ but an accuracy score of just $1/n$.  (Back to the binary case, to compare your method, you'd have to interpret e.g. "Sum(TN)" as including both diagonal entries, so the "accuracy" there is actually $1/2n$, which agrees with the mediant-of-OVR score.)

As such, your metric is similar to macro-averaged scores (though I've never heard of that for accuracy, only precision/recall/Fbeta).  Micro Average vs Macro average Performance in a Multiclass classification setting

Finally, as an opinion, since accuracy measures the probability of getting the prediction right, it's easier to interpret; your metric gives credit for not misclassifying a sample into each class it's not put in, hence the inflation.  Of course, this also perhaps makes a multiclass model's accuracy sound terrible (sklearn says "this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted").