If I recall correctly both of these definitions are equivalent:
- $(E,I)$ is an independence system and satisfies the augmentation property.
- $(E,I)$ is an independence system and all maximal independent sets in I have the same size.
However I would say that the following system satisfies the second definition but not the first:
$(\{1,2,3,4\},\{A|$ all numbers in A have equal parity$\})$
The maximal independent sets are $\{1,3\}$ and $\{2,4\}$, but at the same time if we take $A_1=\{1\}$ and $A_2=\{2,4\}$ we can't find any $a \in \{2,4\} \setminus \{1\}$ so that $(\{1\} \cup a) \in I$.
What am I missing here?
Note: The term "(abstract) simplicial complex" is a common synonym for "independence system", and I'll use the former term here.
Your first definition is one way to define a matroid. Your second definition defines a class of objects called pure simplicial complexes. As you point out in your question, there are pure simplicial complexes that are not matroids.
But what about the converse?
The fact that every basis of a matroid has the same size (equal to the rank of the matroid) follows directly from the augmentation property. It follows that the independent sets of any matroid form a pure simplicial complex. So we have the following chain of strict inclusions:
$$ \mathrm{Matroids} \subsetneq \mathrm{Pure~Simplicial~Complexes} \subsetneq \mathrm{Simplicial~Complexes}$$