Your assumption is a good starting place and its good to be cautious about it. Many of us are guilty of abbreviation or outright misuse of the terms for convenience. Here's my rough guide, not meant to be precise but rather to address the overlapping usage. I've spent a bit more time on the basic definitions as this is probably where variations in usage arises.
Orbital elements: the 6 parameters a, e, RAAN, argument of perigee, and mean anomaly. This describes the shape of an orbit and the position of an object in it, plus a 7th parameter, the time (epoch) at which this situation exists. This is the most general term, all the others in the OP's question are subsets of this term.
Keplerian elements: This specifically refers to the same 6 elements + time that apply to a Keplerian orbit. This means different things according to the context.
- Limited context: orbits that are ideal ellipses. The term thus separates such hypothetical orbits from reality.
- Common usage: the idea of using the six orbital parameters to describe a real orbit, see Osculating below. In this context using the term "Keplerian" could be simply to distinguish an element set as a way of presenting the information rather than presenting the instantaneous position and velocity.
- Much broader definition: referring to item 2 above and distinguishing from a "non-keplerian" orbit, where the latter could refer to an orbit in a 3 body problem or continually under the influence of propulsion or a solar sail.
Osculating elements: the same 6 elements + time but referring to a real orbit, where all the rest of the features of a real orbit such as the non-spherical Earth, atmosphere, solar radiation pressure and the effects of the Sun and the Moon are acknowledged implicitly. Such an orbit is not elliptical, it is lumpy and evolves continually so that the satellite path is different for every revolution.
For this concept to make sense the context has to be that these features are modelled, just not explicitly announced at this moment. Thus the element set might be numerically indistinguishable from a keplerian set (ellipse definition) but would describe a different orbit by virtue of the propagator that is implicitly referenced.
In practice osculating elements are likely to include some additional parameters reference to an object's area/mass ratio in order to support calculation of the effects of atmospheric drag or solar radiation pressure, where the latter will probably roll-in the effects of surface reflectance.
Its a feature of the community that people rarely go out of their way to describe the ins and outs of their model as its assumed as a given. If you wanted to use osculating elements solved by a different satellite operator, for perhaps for co-operation in collision avoidance, it would be up to you to learn about their orbit determination propagator and do comparisons on a mutually known object. I suspect many don't have the budget for such things.
Mean orbital elements: The 6 elements plus time. I have only seen this term in the context of SGP4 TLEs though in principle it could apply to other approaches. There is a public definition of this system and the additional parameters it requires such as for addressing atmospheric drag.
I understand that the lumpiness of the orbit developed from osculating elements is smoothed out. Whilst the name "mean" suggests that this smoothed orbit could be an ideal ellipse, as per Keplerian definition 1 above, the SGP 4 model propagates orbits that are not elliptical, they just have less bumps than modern osculating orbits.
Edit: I found this quote which helps with the "mean elements" concept.
The elements in the two-line element sets are mean elements calculated to fit a set of observations using a specific model—the SGP4/SDP4 orbital model. Just as you shouldn't expect the arithmetic and geometric means of a set of data to have the same value, you shouldn't expect mean elements from different element sets—calculated using different orbital models—to have the same value. The short answer is that you cannot simply reformat the data unless you are willing to accept predictions with unpredictable errors.