What is convertible equity (or a convertible security)?

There is a legal delta between debt and equity as contract classes. What this post is describing is a 'bastardization' of these classes. The net result, is ultimately, an equity contract. It is a "weakened" or "attenuated" option to a convertible note (already, an attenuated alternative to convertible preferred stock). Maybe the OP or PG will chime in with more color.

here is the relevant bits, at least in part

3. Convertible equity is “equity” and probably can be characterized as qualified small business stock, which may have a tax benefit for investors.

4. Convertible debt with a maturity date longer than one year creates problems for California-based investors due to licensing requirements under the California Finance Lenders Law. Making it equity removes this issue.