Definition of $H_\lambda$ (hereditary cardinality)

Although the definition that Trevor quoted from Cantor's Attic fails, as he says, to match the usual meaning of "hereditarily", it is, as far as I know, fairly standard. I suspect the reason (or at least a reason) for that is that it works well in Lévy's cardinal boundedness theorem: Anything $\Delta_1$-definable from sets in $H_\lambda$ is itself in $H_\lambda$. If you change the definition so that a small cofinal subset $A$ of a singular $\lambda$ becomes an element of $H_\lambda$, then this theorem fails, because $\lambda$ itself is $\Delta_1$-definable from $A$ as the union of $A$.

You are correct. For the reasons you discuss, the appropriate definition of $H_\lambda$ for singular $\lambda$ is the set of $x$ such that $x$ and all sets in its transitive closure have size below $\lambda$. (Though I could not instantly find a reference, this is actually standard among those that consider the set at all, which are not too many, and its use is somewhat limited. For example, I would imagine one may want to consider the class in settings where choice fails, but I have yet to see an application in that context.)

On the other hand, it is also standard (and far more common) to only define $H_\lambda$ for $\lambda$ regular.

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Mostly as a curiosity: Some people (Forster, for example), present $H_\kappa$ as $\bigcap\{y\mid \mathcal P_\kappa(y)\subseteq y\}$. This presentation generalizes: We can define $$\mathcal P_\phi(x)=\{y\subseteq x\mid \phi(y)\},$$ so $\mathcal P_\kappa(x)$ is $\mathcal P_{|y|<\kappa}(x)$. One can then define $H_\phi$ as $\bigcap\{y\mid\mathcal P_\phi(y)\subseteq y\}$. This notation seems to go back to

Maurice Boffa. Sur l'ensemble des ensembles héréditairement de puissance inférieure à un cardinal infini donné, Bull. Soc. Math. Belg., 22, (1970), 115–118. MR0309732 (46 #8837).

Note this version coincides with the standard one for $\kappa$ regular, but is the problematic one for $\kappa$ singular.