I'm confused about the terminology in the two contexts since I can't figure out if they have a similar motivation. Afaik, the definitions state that quantum processes should be very slow to be called adiabatic while adiabatic thermodynamic processes are supposed to be those that don't lose heat. Based on my current intuition, this would mean that the thermodynamic process is typically fast (not leaving enough time for heat transfer). What gives, why the apparent mismatch?
The terminological mismatch arises because different physicists use the terms differently in different contexts. For example, here is how Landau and Lifshitz define an adiabatic process in the context of thermodynamics:
Let us suppose that a body is thermally isolated, and is subject to external conditions which vary sufficiently slowly. Such a process is said to be adiabatic
As you can see, these authors combine the criterion of thermal isolation (no heat exchange with the environment) with a slowness assumption, to arrive at their definition of the term adiabatic. In contrast, consider Huang's definition of adiabatic in the context of thermodynamics;
Any transformation the system can undergo in thermal isolation is said to take place adiabatically.
In the context of quantum mechanics, Griffiths defines the term adiabatic as follows:
This gradual change in external conditions characterizes as adiabatic process.
I would say, from personal experience, that the more widely held convention for the term adiabatic is not the one used by Landau and Lifshitz. In particular, most physicists I know use the term adiabatic in the context of thermodynamics to mean thermally isolated, while they use the term adiabatic in the context of quantum mechanics to mean sufficiently slow that certain approximations can be made.
Addendum. In the context of thermodynamics, the free expansion of a thermally isolated ideal gas is often referred to as an "adiabatic free expansion of a gas," see, for example here. Such a process is not isentropic. Using Slavik's definition would deem invalid the characterization of such a free expansion as adiabatic. However, all you need to do is google "adiabatic free expansion" to see how widespread such use of the terminology is.
The etymology of adiabatic appears to be from the Greek meaning "not passable" (native Greek speakers should feel free to clarify and/or correct that). In the technical meanings, the "passing" refers to heat transfer. So in thermodynamics, adiabatic means there is no heat transfer between the system and the environment.
In practice, of course, that's an approximation. In practice, adiabatic means that the thermodynamic process is slow enough that the system is always very nearly in equilibrium, so the heat exchange with the environment is negligible. In quantum mechanics, the analog to equilibrium is an eigenstate. So adiabatic means that the change is so slow that the system is always very nearly in equilibrium, so the system is always in an eigenstate. Both of these are approximations.
In quantum mechanics, the opposite of the adiabatic approximation is the sudden approximation. Take a system with initial Hamiltonian $H_0$, and change the Hamiltonian to $H_1$ over some time $T$. Then the adiabatic approximation is $T \rightarrow \infty$, and the sudden approximation is $T \rightarrow 0$. In the sudden approximation, the state of the system doesn't change (it "doesn't have time to change"), and it finds itself suddenly not in an eigenstate. In the adiabatic approximation, the state follows the perturbation, and is always in an eigenstate of the Hamiltonian.
Adiabatic means quasi-static and isoentropic - slow enough to create negligible amount of irreversible excitation. This is the common rationale of technically different definitions. E.g., Landau & Lifshic'es definition has two components - thermally isolated (to prevent entropy change by heat exchange) and slow (to prevent irreversible excitation). For a gaped quantum system adiabatic can by quite fast (just keep Planck constant times the characteristic driving rate below the value of the energy gap).
What is confusing indeed is that there can be an intermediate speed which you can be reasonably adiabatic - much faster than heat exchange with what you separate as the "reservior" but much slower than the equilibraton speed of the degrees of freedom being excited. That's why adiabatic can be fast and slow at the same time - there are two conditions to satisfy. These subtleties are often not made sufficiently clear,
but that's what we have physics.SE for :)
To sum up, don't make "waves" (entropy) and you'll adiabatic.
In both thermodynamics and quantum mechanics, the term adiabatic is used to refer the idea of "something not changing". In Thermo, this something is that of the system's heat content and is said to not change over the process.
In quantum, this some thing is spoken in terms of "eigen state" of Hamiltonian of a system - precisely, "at the moment of consideration". The notable point here is that, after a time t (non zero), the new quantum system (under adiabatic process) has a different Hamiltonian but the energy of the system at that point is still an eigen value of the new Hamiltonian i.e., the process occurred such a way that system adjusted itself into its eigen states - just like in thermo, system adjusts itself such that its heat content is constant.
In summary, "The state of being an eigen value of Hamiltonian or the heat content (a state of being, in a way)" did not change over the process and hence is referred to as adiabatic.