This is a common point of confusion, not only with regards to inflation, but any time an expanding universe comes up...
The "cosmic speed limit" as you call it says that no particle or signal can move through spacetime faster than the speed of light. Spacetime is a very specifically defined thing, described with a coordinate system. There is no restriction, in terms of speed, on what spacetime itself is allowed to do. Let me illustrate with an example.
Imagine a photon. Relativity tells us that it always travels at speed $c$ (exactly at the speed limit). Let's say the photon has a path 10 light years long to travel along (remember light years are a measure of distance, $1\mathrm{ly} =$ the distance travelled by a photon in 1 year). The photon leaves and travels for 5 years, covering a distance of 5 light years. Then very suddenly, the universe doubles in size! The photon continues on its journey. The 5 remaining light years to travel have doubled in size, so it travels 10 more years to cover the last 10 light years. The journey has lasted 15 years. But the photon is now 20 light years from its starting point. Naively, we might compute its speed as $v = 20\mathrm{ly}/15\mathrm{yr} = \frac{4}{3}c$, faster than the speed of light. But in reality, it was just moving at speed $c$ the whole time in a universe that expanded.
In a more realistic scenario, the universe doesn't "suddenly" double in size, it does it gradually, but conceptually the same thing happens... you just need to use integrals to work out the math.
As to the meaning of time, that's somewhat more philosophical. However, I'll point out that, at least in general relativity, time is on an equal footing with space. Spacetime is described by a mathematical object called a metric. One example of a metric looks like:
$$ds^2 = c^2dt^2-dx^2-dy^2-dz^2$$
$x,y,z$ are the spatial coordinates and $t$ is the time coordinate; $s$ is a sort of generalized measure of spacetime length. As you can see, other than the constant $c$ (which could be set to equal 1 with a clever choice of units, so it's really rather unimportant), and a negative sign, time and space are equivalent in this formalism. If you understand space, then time should also make sense, as it's simply related to space by your "cosmic speed limit".
The Big Bang was originally defined as the zero time limit of the FLRW metric, so it's a mathematical construct and not primarily something physical. We have chosen to apply it to the zero time limit of the universe because we thought the FLRW metric was a good description of the universe, but then inflation gatecrashed the party and spoiled the fun.
So if you're going to use the phrase Big Bang in connection with the universe, as opposed to its purely mathematical meaning, then it's up to you to define what it means. As you've found, there is currently no consensus on its meaning.
Personally I would avoid using the term unless you're specifically referring to the FLRW metric.
Best Answer
The Planck era is defined as the time when the universe was the size of the Planck length, $10^{-33}$ cms, and less, and the universe's age was $10^{-43}$ sec, the Planck time, and less. It is the earliest epoch we identify after the Big Bang. The Planck temperature at the end of the epoch was about $10^{32}$ degrees Kelvin.
This was way before quarks, leptons, Higgs bosons, and inflation. Neither Quantum Theory (QT) nor General Relativity (GR) have anything to say about what is happening at these sizes, times, energy densities and temperatures, except that they are not applicable.
The wiki article summarizes the epochs, or times, when the universe was dominated by different kinds of physics, from the Planck epoch through the GUT epoch, inflation, electroweak and strong force separation, and on till the current epoch. It is at https://en.m.wikipedia.org/wiki/Chronology_of_the_universe
At Planck times and earlier after the Big Bang, GR effects or equivalently strong gravity effects predicted by GR, would be happening at the sizes and energies such that QT also applies. It needs a unification of GR and QT to have anY explanatory power, a theory of quantum gravity. At these times there are no photons, quarks, electrons, neutrinos, gluons or anything associated specifically with the 4 forces. This was before spacetime was defined by geometry and GR, or even before it existed. The two best known theories of quantum gravity, none proven or accepted, are string theory (ST) and its newer versions of superstrings and M theory, and loop quantum gravity. They both have elementary entities which can be about the Planck size, one is strings and the other loops.
I know quantum loop gravity less well, but when the spacetime is larger than the Planck length ST has strings in a 10 or 11 dimensional spacetime, which were all small dimensions unti the inflationary era where for some reason all but 4 dimensions remained small, and so now we see those 4 dimensions. At least that is one version. Others take multidimensional branes in a higher dimensional space. You can see the wiki references on string theory. But either way at the Planck length it becomes a sort of quantum foam, with no spacetime defined. Spacetime emerges as the scale factor increases and multiple Planck lengths enter in.
So, what happens at the Planck epoch (era) is not well understood. As the universe goes into the next era, the GUT era, 3 of the 4 forces, all but gravity which when we enter the GUT epoch decouples, are unified. ST aims to also explain it, i.e., a unification of the 3 forces (and actually gravity in the Planck era as well). The 3 forces then remain unified until the universe expands and cools some more, the strong force decouples, and later there is inflation and the electroweak force separates into the weak and electromagnetic force, and the Higgs boson emerges. There is more after that.
See the wiki and its references for string theory cosmology, but at inflation and later (maybe even before, up to the Planck epoch) there is commonality with the standard cosmology model. https://en.m.wikipedia.org/wiki/String_cosmology