The only reason behind the existence of the Special Theory of Relativity is the experimental fact that the speed of light in a vacuum is independent of any motion of the source or receiver. This is because the usual concept of speed is inconsistent with the invariance of c unless one redefines length and time units accordingly.
In his paper 'On the Electrodynamics of Moving Bodies' (published in 1905), Einstein derives his theory by considering the propagation of a light signal in two different reference frames, each of which being defined by its own set of detectors (stopwatches) that should serve to define the locations (and thus the speed of light) in each frame. Two different sets of detectors would indeed by necessary if the speed of light is invariant in each reference frame, but Einstein then suddenly makes the assumption that he doesn't need the second set of detectors, but can obtain the location in the primed frame by using the location in the unprimed frame and projecting this into the primed frame by means of the Galilei transformation x'=xvt (note that here x' is not the variable x' in Einstein's 1905 paper but the transformed coordinate (which he denotes as ξ)). This however corresponds to the usual concept of 'speed' which can not be applied to light (this is for instance immediately apparent from the example in the box below). Half aware that this obviously would lead to an inconsistency with the principle of the invariance of c, he then decides to multiply the right hand side of the equation by an adhoc factor γ, i.e. x'=γ
^{.}(xvt), such that the equation is formally valid again. The rest of his derivation is then devoted to determining this adhoc factor γ for this so called 'Lorentz transformation'. It should be obvious however that this procedure is no less mathematically flawed than having for instance the inconsistent 'equation' 1=2, multiplying the right hand side by a factor k , i.e. 1=k
^{.}2, and then saying that the equation is consistent for k=1/2 (in Einstein's case it becomes indeed 'consistent' for γ= 1/
√[1(v/c)
^{2}] (after applying a further adhoc correction to the time t to complete his 'transformation')). It should be clear that such a redefinition of the original length and time units in is not only mathematically but also physically inconsistent as it essentially means that there is no basis for any measurement anymore (it is equivalent to using your own customized measuring stick when, let's say, a certain job requires you to be 180cm tall but you measure only 150cm; you can rescale your measuring units so that it shows 180cm, but it is obvious that this is an incorrect and illegal procedure).
I have given a stricter mathematical analysis of Einstein's 1905 derivation on my page
Mathematical Flaws in Einstein's 'On the Electrodynamics of Moving Bodies'.
Furthermore, I have also pointed out the inconsistencies in his later
simple algebraic derivation which he published in his book 'Relativity: The Special and General Theory'.
It should be clear from all this that the conceptual and mathematical inconsistencies are not tied to a particular derivation of the Lorentz transformation, but are inherent to the logical contradiction of trying to apply the usual concept of speed (which implies that it is frame dependent) to light (which has a frameindependent 'speed') (see the page
Speed of Light Postulate and Lorentz Transformation for more).
To clarify the meaning of the invariance of the speed of light and the velocity dependence in the 'transformation' equation, consider first the propagation of a massive body e.g. a bullet fired from a gun (in the absence of any external forces). Assume you have a number of detectors distributed along the path of the bullet, each associated with a synchronized clock that is stopped when the bullet passes the detector at the particular location. Define furthermore the location of the first detector as x=0 and the time any other detector shows as the difference time t to this detector.
Now, as case (a), assume first the gun is resting with regard to the detectors when the bullet is fired. If you then go and have a look at what time each detector at a particular distance x is stopped, you will find the relationship x=ut, where u is the velocity of the bullet relative to the gun.
Then, as case (b), assume the gun is receding from the detectors with velocity v. Again, if you go and have a look at what time the detector clocks at a particular distance were stopped, you will find the relationship x'=(uv) ^{.}t (where the primed quantity shall indicate that the corresponding location is associated with the moving gun).
Now do the same experiment not with a gun firing a bullet but a light pulse.
Again, as case (a), assume first that the light gun is resting with regard to the detectors. If you check the clock times at the detectors, you will find the relationship x=ct (with c the speed of light).
Now the question is what relationship between the locations and trigger times of the detectors will you observe if the light gun is receding from the detectors with velocity v (case(b))? This is were the principle of the invariance of c comes in. Let's assume that such a principle exists, which means that the trigger time at a given detector must be the same as in case (a), i.e. trivially we have x'=x=ct. So it is obvious that not only is there no further 'transformation' needed to find the coordinates of a light signal in case of a moving light source, but it would indeed logically contradict the invariance principle (which by itself provides already the coordinates).
Of course, both the 'massive particle' and the light case are completely separate and are incompatible with each other (x'=(uv) ^{.}t is incompatible with x'=ct, as otherwise one would have uv=c, which would contradict the invariance of c ).

The mental picture of light as an entity travelling independently through space is therefore wrong. If one wants c to be invariant, there can only be a moment of emission and a moment of detection, and the 'speed' of light is simply given by the difference between the two and the distance at the instant of emission (i.e. T=x(t=0)/c ). One might ask how the light signal 'knows' what this distance is, but this would be a metaphysical question like the one how the earth knows how to react to the gravitational pull of the sun. It is just a law of nature.
This circumstance could for instance be of relevance for the apparent
anomalous acceleration (slowdown) of the Pioneer 10 and 11 spacecrafts, as the traditional way of calculating the travel time of the communication signal underestimates the corresponding distance approximately by the observed mismatch (it is obvious that for the speeds involved (10s of km/sec) this problem will only become apparent for large distances, as observed).
A concluding word regarding the invariance of c: this is generally displayed merely as an experimental fact, but if one accepts that light is an electromagnetic wave, it is indeed a theoretical necessity: a lightwave can not possibly require a physical carrier medium because otherwise it could not propagate through empty space; on the other hand, without an 'ether', the speed of light has to be constant with regard to source and receiver or otherwise it would be completely undefined. The point is in fact that a light wave needs no carrying medium as it carries itself (to be more precise, according to Maxwell's Equations, the electric wave carries the magnetic wave and vice versa; it is somewhat ironic that Maxwell himself did obviously not realize this as he believed in the ether theory and a positive outcome of the MichelsonMorley experiment). On the other hand, experiments like the Sagnac effect, which apparently contradict the invariance of c and suggest the presence of an absolute reference frame, could well be explained by the presence of the earth's magnetic field. One should expect that with a sufficiently compensated magnetic field these experiments also give a negative result i.e. the conclusions on this page should then apply.
Of course, for the addition of velocities of material objects, the speed of light is not relevant at all and velocities are added in the usual way, i.e. the relative speed between two objects can be larger than c.
NOTE: Because the usual addition of velocities does not apply to light, the notion of a 'speed' or 'velocity' of light should therefore be used only with care. With regard to light there can strictly speaking only be distances and light travel times (at least as the propagation in a field free vacuum is concerned).