It's a perfectly valid interpretation. If you have a look at the question How long would it take me to travel to a distant star? you'll find it's quite possible to cover a distance greater than $ct$ in an elapsed time $t$. But this doesn't mean special relativity is wrong - indeed the calculations done in that Q/A were done using special relativity.
The problem is that when teaching students special relativity we start by using simplifications that apply only to inertial frames i.e. situations where there is no acceleration (or indeed spacetime curvature, as in GR). Students then apply these simplifications to situations where there is acceleration, and as a result run into problems like the apparent FTL travel you describe, or most notoriously the twin paradox.
To understand what's going on you need to know that the invariance of the speed is light is a local property. That is, if you measure the speed of light at your position you'll get the value $c$, and if you measure the speed of any massive object at your position you'll get a value less than $c$.
If a measurement is not local neither of these results is guaranteed. For example right now there are distant galaxies moving at faster than light relative to us. This is a result derived using general relativity, but a related effect happens even in special relativity. If you accelerate at a constant acceleration for long enough a horizon will develop behind you (the Rindler horizon), and the horizon develops because everything behind it is travelling faster than light relative to you.
So you're quite correct that superluminal travel is possible in the sense you describe, but superluminal travel is still impossible in the sense Einstein described.