I would say that if you are looking for a concrete definition then it's better to adopt the Tannakian point of view and to focus on the category of representations of the quantum group rather than on the algebra itself. So take as your fundamental object a tensor category (a special type of rigid abelian monoidal category - see here for details). A "quantum group" is then some way of realising the category as a category of representations or corepresentations. There can be different algebras which do this job, and they can come in different flavours, such as Hopf algebras or compact quantum groups in the sense of Woronowicz. This allows one to view the various quantum groups floating around as tools to study the category itself, removing the need for any axiomatic definition.
Finally, if it's really deserving of the name "quantum group" then the category should admit a braiding. For example, this will give a quasi-triangular structure (or dually a coquasi-triangular structure) for any Hopf algebra realising it (see the comment by Sam Hopkins above).