You mention that you are also interested in the satellite-orbiting-a-celestial-body scenario.
There is a discussion written by Kevin Brown, in 1997, posted in the Usenet group sci.physics
Nowadays Googlegroups is the custodian of the Usenet archive.
The name of the thread is Relativistic time on satellites
Scroll to the answer posted by Kevin Brown.
(The content has suffered a bit; Usenet communication is written to be displayed with a fixed width font, such as New Courier. Formula's are sometimes formatted on three lines, using spaces for alignment. To read the Usenet message: copy the text, paste it in a text editor, and set the font to a fixed width font.)(later edit: I'm not sure, but I think content has been damaged; I think leading spaces have been removed.)
Kevin Brown starts with pointing out that in terms of General Relativity there is is only a single time dilation. In orbital motion there is a single time dilation. Distinction between velocity time dilation and gravitational time dilation is a matter of interpretation; it's not inherent in GR.
Separation in velocity time dilation component and gravitational time dilation component can be obtained for example as follows: compute the total time dilation for orbit at some altitude, then compare that to gravitational time dilation between surface of the planet and being stationary at that altitude.
For a satellite in low Earth orbit, altitude about 300 kilometers, the difference in altitude with the Earth surface is negligable. (Gravitational potential is with respect to the Earth center.) In low Earth orbit the orbital velocity is very large, one circumnavigation about every 90 minutes. So: for a satellite orbiting in low Earth orbit a smaller amount of proper time elapses than on the Earth's surface.
As we know, there is an altitude where one circumnavigation of the Earth takes a sidereal day, giving rise to geostationary orbit. Given that gravity is an inverse square force: the higher the altitude of the orbiting motion the slower the velocity. It's not just the angular velocity that is slower, the tangential velocity is slower too.
At geostationary orbit altitude the difference in gravitational potential is such that for a satellite in geostationary orbit a larger amount of proper time elapses than the amount of proper time that elapses at Earth surfacee.
The amount of proper time that elapses for a satellite in orbit goes from smaller in low Earth orbit to larger in geostationary orbit.
Kevin Brown derives a formula for relativistic time on a satellite, finding that the cross-over altitude is at about 2000 miles up. So if you would place a satellite at precisely that orbital altitude the amount of proper time elapsing for that satellite would be the same as at Earth surface.
Incidentally, In your question you phrased everything in terms of 'observer A sees' and 'observer B sees'.
In the case of orbiting motion the effect is cumulative, in which case it is meaningful (and in my opinion better) to phrase the time effect in terms of 'amount of proper time that elapses'
Proper time is a well defined concept. Muons have half-life of 1.56 micro seconds. But for Muons moving at relativistic velocity in a storage ring a smaller amount of proper time elapses per unit of laboratory proper time. So the Muons in the storage ring are available for the experiment for a longer time (laboratory proper time).
Proper time is well defined in the following sense:
You create a scenario with protagonists who are all experts in applying relativistic physics. Being experts they will all agree on the amount of proper time that wil elapse onboard a satellite orbiting at a particular altitude. They will agree on the ratio of proper time onboard the satellite compared to the proper time at Earth surface.