The quick answer: It's complicated and depends on latitudinal domain of interest or your motivation for caring about how the ionosphere is defined. What's important to an HF person is different from that for a UHF person or VLF person.  

Generally, the upper boundary of the ionosphere is taken to be the O+/H+ transition height, above which the dominant species is H+. The H+ -dominated region is generally referred to as the protonosphere or plasmasphere and is referred to separately from the ionosphere. There are dozens of papers out there about the O+/H+ transition height, as it is characterized by a change in the dominant scale height and is thereby incredibly important for electron density modelling. Depending again on your motivation for caring about the ionospheric domain, some people include the plasmasphere in their definition of the ionosphere. For example, a GPS person may define the ionosphere as the region of the earth's upper atmosphere that contains an appreciably dense plasma in the context of how that plasma affects GPS signals.

For the purpose of systems like GNSS, the plasmasphere should be taken into consideration when doing reconstruction with GNSS-based TEC. To effectively reconstruct electron density with these systems, the location/altitude of the plasmapause is extremely important, as this outlines the outermost limit of the necessary reconstruction domain. This plasmapause is however dynamic and depends strongly on latitude. For example, at high latitudes, like say the polar cap region, the plasmasphere is virtually non-existent and plasmaspheric electron content can be considered a negligible contributor to Total Electron Content (TEC).

On the other end of the altitude spectrum, the D-region is generally negligible in terms of electron density/content; however, the high collision frequency at those altitudes makes it an incredibly important contributor to signal absorption and should thereby always come into consideration in HF propagation models (the very common habit of neglecting collisions in refraction modeling can lead to significant errors at HF frequencies). Below 80km, however, is generally considered part of the mesosphere and not the ionosphere.

As for your a priori guess for hmF2, it depends again on latitude. When in doubt, use a model rather than a 300km or 350km guess. Both of those guesses would be horrible in the equatorial anomaly region but may be fairly apt at mid latitudes. You can always justify yourself at your defense through the use of a model (however that brings up the next question of why you chose that specific model but that's life, lol).

The quick answer: It's complicated and depends on latitudinal domain of interest or your motivation for caring about how the ionosphere is defined. What's important to an HF person is different from that for a UHF person or VLF person.  

Generally, the upper boundary of the ionosphere is taken to be the O+/H+ transition height, above which the dominant species is H+. The H+ -dominated region is generally referred to as the protonosphere or plasmasphere and is referred to separately from the ionosphere. There are dozens of papers out there about the O+/H+ transition height, as it is characterized by a change in the dominant scale height and is thereby incredibly important for electron density modelling. Depending again on your motivation for caring about the ionospheric domain, some people include the plasmasphere in their definition of the ionosphere. For example, a GPS person may define the ionosphere as the region of the earth's upper atmosphere that contains an appreciably dense plasma in the context of how that plasma affects GPS signals.

For the purpose of systems like GNSS, the plasmasphere should be taken into consideration when doing reconstruction with GNSS-based TEC. To effectively reconstruct electron density with these systems, the location/altitude of the plasmapause is extremely important, as this outlines the outermost limit of the necessary reconstruction domain. This plasmapause is however dynamic and depends strongly on latitude. For example, at high latitudes, like say the polar cap region, the plasmasphere is virtually non-existent and plasmaspheric electron content can be considered a negligible contributor to Total Electron Content (TEC).

On the other end of the altitude spectrum, the D-region is generally negligible in terms of electron density/content; however, the high collision frequency at those altitudes makes it an incredibly important contributor to signal absorption and should thereby always come into consideration in HF propagation models (the very common habit of neglecting collisions in refraction modeling can lead to significant errors at HF frequencies). Below 80km, however, is generally considered part of the mesosphere and not the ionosphere.

As for your a priori guess for hmF2, it depends again on latitude. When in doubt, use a model rather than a 300km or 350km guess. Both of those guesses would be horrible in the equatorial anomaly region but may be fairly apt at mid latitudes. You can always justify yourself at your defense through the use of a model (however that brings up the next question of why you chose that specific model but that's life, lol).

Look, here are quite distinct borders which could be determined physically, and then formally. What is main feature distinguishing ionosphere from atmosphere - its electromagnetic properties due to ionization. Regarding the lower border we should consider electric filed penetration into ionosphere. It could be presented as vertical only to the altitude 60 km, at higher altitudes you should take into account the anisotropy of conductivity, so 60 km we can consider as the lower border having the physical reason for substantiation. Next, what is the difference between the plasmasphere and ionosphere? Plasmasphere is fully ionized and ionosphere is partly ionized atmosphere. In F-region of ionosphere the concentration of neutral gases still 5 orders of magnitude higher than ionized species. So, we can determine by the upper border of the ionosphere not a specific altitude 1000 km or 2000 km, but the altitude where the concentration of neutrals and ionized species at least are equal or ionized part is 75% (it is a matter of consensus). But such determination will be more appropriate from the physical point of view.

Are you sure that the ionosphere is spherical with a fixed radius? It has already been pointed out that the height varies with latitude, but I wonder whether it does not also vary with time of day. Is the ionosphere facing the sun the same height as that in darkness? Do lunar and solar tides affect its height? And what happens to its height when subjected to solar flares, etc?

Hope this helps.

Thank you David, Sergey and Alastair for your high quality responses. This is exactly the kind of discussion I was hoping to start with this question.