tl;dr: milli-G and micro-G refer to literal, specific-ish levels of acceleration. "Microgravity" is a general term for freefall which is not used synonymously with "micro-G." There is no evidence in the literature below that intention has anything to do with referring to "milli-G" or "micro-G".
The only non-opinion-based way I can think to answer is with a literature review. Doing a proper one would probably result in an actual academic paper. I am going to do a half-assed one instead.
I'm starting with a Google Scholar search, with the prompt "milli-g space".
Multi-parameter rover wheel and grouser optimization for deployment in Phobos' milli-g environment is using "milli-g" for accelerations of some tenths of milli-Gs: per the paper, Phobos' gravity corresponds to between roughly 300 and 700 micro-G (see quote below). The term "microgravity" does not appear in the paper.
Effective gravity on Phobos is very low and
fluctuates roughly between 3 × 10-3 m/s² and
6.8 × 10-3 m/s² . However, due to the probable
landing site choice, gravitational acceleration of over
5 × 10-3 m/s² has to be expected .
Microgravity acceleration measurement system for the International Space Station uses milli-G as some specific range as distinguished from nano-G, so likely very literally some thousandths of a G acceleration. I can't read the whole paper but here's an excerpt from the abstract:
The MAMS instrument will provide highly accurate acceleration measurement data over the nano-g to milli-g range characterizing the Lab Module environment in the frequency spectrum from 10/sup -4/ Hz to 100 Hz.
And an excerpt from an explainer page on Glenn Research Center's servers:
The MAMS is a complimentary acceleration measurement system to SAMS. While the SAMS system measures acceleration disturbances from 0.01 to 400 Hz, the MAMS measures accelerations from DC to a maximum of 0.01 Hz. MAMS provides this complimentary function by measuring accelerations caused by the aerodynamic drag of the ISS as it orbits the earth. In addition, MAMS measures accelerations caused by small ISS attitude adjustments and ISS gravity gradient effects.
Active vibration isolation of ultra-stable optical reference cavity of space optical clock also uses both micro-G and milli-G to refer specifically to a certain magnitude of acceleration:
Inevitably, the micro-vibration level in the order of micro-g (μg) is required for ultra-stable optical reference cavity of space optical clocks. The realization of such a strict micro-vibration requirement over broadband frequencies is still technically challenging under complex space disturbances in the order of milli-g (mg). This paper thus proposes a multi-degree of freedom (DOF) active vibration isolation approach for the ultra-stable optical reference cavity to satisfy the stringent micro-vibration requirement.
Elsewhere in the paper "microgravity" is used more generally and informally:
The Suppression of Transient Accelerations By Levitation (STABLE) system was the first successful flight test of an active isolation device for microgravity science payloads using non-contact electromagnetic Lorentz force actuators 
Inertially stabilized two-axis gimbal for space laser communication systems: design description and test results I also can't get to, but here's the quote Google gave me:
of 1 milli-G, RMS, (simulating a worst case disturbance environment anticipated for space
This also seems to refer specifically to literally one-thousandth of a G of acceleration.
Based on this small, somewhat random sampling (the space-related results Google Scholar gave me on the first page of the search above), "milli-g" and "micro-g" (with inconsistent capitalization of G; I prefer capital, but have not edited the quotes above) are distinct terms that refer to actual levels of acceleration, though the Phobos paper demonstrates that the "milli-G" range is probably a couple of orders of magnitude broad. Microgravity is a distinct term referring more generally to freefall / space environments; I admittedly did not search to see if there is some other distinct term for freefall environments with more-severe acceleration perturbations.
If someone else would like to extend this possibly-already-too-long answer (or finds literature that supports a different conclusion than I have come to), I am more than open to turning it into a Community Wiki. Also, sorry/you're welcome to uhoh--I don't know you super well, but I know that giving you papers can send you off on academic journeys, so I hope these are good ones.