I had the same question, and have found a simple answer that satisfies me. While I may be late to the party, let me give this one a try.
The answer is pretty easy, really. Starting from the basics, coupling means that two oscillators are connected in some way that allows them to affect each other's motion, which ultimately leads to exchange of energy. What matters next is whether the coupling comes with energy dissipation/loss.
The "normal" coupling we all learn in rudimentary mechanics is what you call reactive coupling here: there is NO loss of energy from this phenomenon. It is also usually called coherent coupling in some systems, for example here. This makes the system conserved.
Meanwhile, dissipative coupling comes with energy loss, which makes the system, well, dissipative.
It's as simple as that!
While it seems like your question is concerned with a quantum oscillation, thinking of the oscillation classically is useful, as it is just a quantization away from the quantum version—though said quantization is not always so simple (for example here). I find it helpful to think about two coupled pendulums.
When the two pendulums are coupled, we can imagine reactive coupling as the two pendulums connected by an ideal spring. One pendulum affects the other through this spring, transferring energy in the process without losing any of it. Meanwhile, dissipative coupling can be visualized as a damping piston connecting the two pendulums. One pendulum affects the other through the piston, but some energy is lost in the process due to the damping present in the coupling. See this article for a nice visualization.
If you would like to speak more mathematically of the quantum formulation, I shall do it in this paragraph. Firstly, you are correct that reactive coupling always appears in the Hamiltonian, but the term must be Hermitian to conserve energy (have a read). On the other hand, the mathematical description of dissipative coupling need not use a Lindblad dissipator—the Lindblad master equation is not the only way to describe a lossy system, after all. You can also describe it using a non-Hermitian term in the Hamiltonian, like is done here.
Answering the big question: how do we interpret all of this?
Reactive coupling is two oscillating quantum systems (not necessarily mechanical) exchanging energy with each other through some kind of interaction, in which no energy is loss thus keeping the total energy of the combined system constant.
Dissipative coupling is two oscillating quantum systems (again, not necessarily mechanical) exchanging energy with each other through some kind of interaction, in which energy is loss to the environment. The "environment" here is anything that is not the two oscillating quantum systems we care about (surely you already understand this). So, no, it's not an exchange of energy through the environment. Once energy is lost to the environment, it is lost for good. Read this cool book.
This is not all there is to learn about reactive and dissipative coupling. For example, interesting physics arises when we have both types of coupling in the system. Depending on the relative strength of these couplings, cool things might happen. Read here for more.
Some more interesting reads: here and here.
