Solidly grounded means that you have a direct connection between the grounded polarity, and the EGC. For separately-derived systems in general, you also connect the GEC at the location where you bond the grounded conductor and EGC. The word "solidly" means the continuity is intended to be permanent between the groundED conductor and groundING conductor, and that there is no reason for the continuity to ever be removed.
Functionally grounded means that the system has a reference to ground in some form or another, but with an indirect connection. Like a connection through power electronics, or a connection through a protective device. The NEC has an informational note that indicates examples of what kinds of systems this would be. The short answer, is that it is most systems you'd build today would count as functionally grounded.
One example is what we used to just call a grounded system, which bonded one of the polarities to ground via a GFCI fuse or breaker. During normal circumstances, the grounded polarity would have continuity to the EGC, exclusively through this GFCI device. When there is a ground fault, this device would trip, and temporarily make both polarities ungrounded, until the ground fault is fixed. Such systems had an isolation transformer as part of the inverter, because the waveform produced by the power electronics would have a DC offset that needed to be eliminated to in order to be connected to a grounded AC grid. The indirect grounding of this kind of system was necessary with the way the required GFCI technology worked at the time.
Another example of what is now called a functionally-grounded system, is what non-isolated inverters typically use. We used to call this an ungrounded system. Instead of using a GFCI fuse or breaker, this kind of system uses a residual current sensor that looks for discrepancies between the positive and negative current, in order to identify a ground fault. These inverters have both polarities ungrounded, but by the nature of being connected to a grounded AC grid without isolation, the two polarities are at equal and opposite voltages when measured to ground. Suppose there is 500V between the polarities, this would mean that the positive is at +250V, and the negative is at -250V. This DC setup could therefore generate both the positive half, and the negative half of the AC waveform, and have it be symmetric about ground, as it needs to be in order to feed in to the AC grid. This kind of inverter will make a functionally-grounded DC system, when connected to a grounded AC grid.
A truly ungrounded system would be created, if you connect a non-isolated inverter to an ungrounded AC grid.