A lady with a coat of arms was almost certainly one who had a seal that was used for land transactions and thus owned land. A lady with a title was almost certainly a tenant of large tracts of land from the king. Such ladies were therefore indeed heiresses as in the modern use of the word.

But as time wore on, the rights of inheritance became stronger and the rights of the king to feudal powers over his tenants became less. So a lady could indeed inherit, or at least pass on, a coat of arms and she could indeed inherit a title but there did not have to be any wealth attached to either.

So we have heiresses who are indeed heirs to important "property" but not to any wealth. They are just as much heiresses as the monied ones but to distinguish them, it is convenient to call them "armorial heiresses".

I have preferred to call them "genetic heiresses" as this described their primary condition, that they either had no brothers or that their brothers' lines died out. There may be some genetic reason behind this, particularly when one also sees so many other families whose lines never die out.

So a genetic heiress may not have money and she may not inherit a coat of arms or a title. Money does not have to come into this study.