Part of me thinks that if the LSZ formula and asymptotic states (see here) were discussed rigorously in introductory QFT books, instead of the sloppy presentation that is usually given (if at all), all this confusion around virtual particles would be greatly reduced.
QFT is just infinite dimensional QM, and in QM, all you have are states and a Hamiltonian. Nothing more. There's nowhere for "virtual" states to hide.
A photon that travels from alpha centauri to your eye is a real photon, end of story. The little wavepacket that the photon that travels freely through space corresponds to a state which is recognizable in the Hilbert space.
I invite you to read closely the blog post of Matt Strassler you posted, because I strongly agree with the view he advances.
In reading it just now, it occured to me that, in some sense, the difference between a particle and a virtual particle has nothing to do with quantum mechanics at all, but is even present in classical physics. Take for instance Maxwell E&M. The Coulomb $1/r^2$ field surrounding a charge is what is directly responsible to the attraction of charges over a distance. However, there are no light waves in this solution. Light waves are created when the charge is wiggled. This creates a propagating $1/r$ field. Griffiths in his E&M textbook gives a nice little analogy, saying that the $1/r^2$ Coulomb field is like the flies buzzing around near a garbage truck, but if a group of those flies splits off and flies off in some random direction, that's radiation, A.K.A. "light."
In classical Maxwell E&M, all that exists are the $E$ and $B$ field. Some of those solutions we identify with light waves (little propagating sinusoidal waves) but NOT all solutions! There are no light waves present in the $1/r^2$ Coulomb potential solution, and THIS is the solution mainly responsible for the attraction of charges.
Just the $E$ and $B$ fields are all that exists in classical E&M, the Hilbert space is all that exists in QFT. Sometimes a state in the Hilbert space has a natural interpretation as being made of particle states, like a photon emitted from alpha centauri moving across the cosmos, but not every state is understandable in terms of particles. Roughly speaking, I think the following analogy is a pretty good one:
Particle : Virtual particle = Light wave : Coulomb Field.
(However, don't get toooo excited about this analogy. As ACuriousMind comments, 'Strassler is playing a bit of a word game and uses "virtual particle" as synonymous with "intermediate state that's not a pure particle" in a manner different from "internal line in a Feynman diagram".' I think this is also a fair take, and really gets more into the language of what the heck a virtual particle is anyway.)
Also @Deschele Schilder's answer seems to disagree with yours somewhat, how you would respond to the difference?
One must remember exactly when the LSZ formula applies, because that's when all this scattering and Feynman diagram stuff comes into play. LSZ is relevant when you have widely separated wave packets which come to overlap significantly over some spatial region called the 'interaction zone' (see the answer and picture here). Now, when alpha centauri releases a photon, a widely separated wave packet leaving the star is exactly what you get. In other words, the photon leaving alpha centauri is already the `external leg' of the process that created it.
Also, philosophically, why should the electron in your eye be any more real than the photon that excited it? Who's to say that electrons being excited are "real" measurement, and the photons are just virtual? Couldn't you say that the only way you could ever detect electrons is by the emission and absorption of photons? That seems to me to produce a rather vile game of chicken and egg.
Would I be right in saying if you were wanting to calculate the interaction between an electron in Alpha Centauri and an electron in your eye you would have 'virtual photons' in the QED calculation, but that this is calculating a very separate process to the real photon absorption process in my question?
That's a good question, and I don't actually think you would be correct in saying that. For one, the electron in your eye is bound in an atom, in its own orbital, and is not a free electron. So really the correct Feynman diagram would probably involve an atom absorbing a photon and raising into an excited state. There's also a wide range of stuff floating out in the cosmos which you have to account for, so I would also imagine that decoherence effects would play a big effect on the photon on its journey.
To my mind, the most technically correct treatment would be to have the photon emitted as the external leg of some emission process on alpha centauri, then let it travel as an honest to god particle, and then draw a separate Feynman diagram for the absorption process by the atom. I think such a treatment has a stronger basis in the actual LSZ formula, which is where all of this particle scattering stuff is coming from when you get down to it.
It is crazy how I seem to see these sort of ideas spread by people with quite a large physics educational background through.
I think the fact that physicists with a high level of education disagree strongly on matter of interpretation and language becomes less crazy the more highly educated physicists you meet.