Can neutrinos interact by the EM interaction and gravity? A definition of a lepton is:

A particle that does not interact by the strong force but does by the 3 other forces.$^1$

Neutrinos are leptons, so from the above definition one would expect it to interact through the weak interactions, EM and gravity. However I have seen arguments like the following:

This interaction must be weak since it contains neutrinos.$^2$

Clearly this implies that neutrinos only interact through the weak interaction and thus contradicts the definition given above.
So can neutrinos actually interact through EM and gravity? If so why do you often get arguments like the one in Martin and Shaw?

$^1$Adapted from: Matolyak, J. and Haija, A. 2014. Essential Physics. Boca Raton: CRC Press, page 442
$^2$e.g. Martin, B.R. and Shaw, G., 2013. Particle physics. John Wiley & Sons. page 337
 A: A neutrino is thought to interact only through the weak force and gravity. They interact primarily, though, through the weak force (perhaps explaining the Martin/Shaw comment). 
Interestingly, since the neutrino has a minuscule mass (as opposed to none at all), it could have tiny neutrino magnetic movements, therefore allowing the possibility that it could interact through EM as well. There is a possibility of sterile neutrinos (a fourth type suggested by experimental data from the LSND experiment); these would not participate in weak-force interactions. 
Antineutrinos interact with matter through only the gravitational and weak forces, therefore making them difficult to detect experimentally. Concerning neutrinos, I might add that if you go to this website, there is more information about this.
Finally, interestingly enough, I looked at the Wikipedia page for leptons (here). It simply defines a lepton as a spin $\frac{1}{2}$ particle that does not undergo strong interactions. It further states that leptons are divided into two classes: charged leptons (also known as electron-like leptons) and neutral leptons (aka neutrinos). 
Hope this helps!
