Is photon truly a photon only in vacuum? To me, it seems like photon becomes “heavier” in different mediums, as its speed slows down. And, as I know, only particles with mass travel slower than the speed of light.
 A: One should not confuse light with photons, in the same way that one does not confuse the ripples in a pond with the molecules of water. Ripples and molecules have different physical behaviors and obey different boundary conditions, nevertheless a stone thrown in the pond will generate ripples, and molecules will move back and forth to pass the ripple on, similar to the wave in a stadium.  where people are not the wave.
Light emerges in a mathematically complicated way riding on zillions of photons . Photons obey a quantum mechanical Maxwell's equation and thus the emergence of collective effects as light is explained. In this image pictorially for a special case one can get an intuition of  how light emerges from photons.


Left and right circular polarization and their associate angular momenta

The photons, in the middle, have only spin + or - in their direction of motion. They build up the polarization of the wave.
With this as physics background:

To me, it seems like photon becomes “heavier” in different mediums, as its speed slows down. And, as I know, only particles with mass travel slower than the speed of light.

It is not the photon that becomes heavier, it is light as explained in the other answer. The photons in a transparent medium travel longer paths due to quantum mechanical  elastic scatterings with the lattice of the medium and still keep coherence so as to transfer images. Longer paths explain how their velocity is still c while the emergent light slows down.
Due to the angles introduced by the elastic scattering of the constituent photons in transparent media, an invariant mass for the light, which is a conglomerate of photons,  is generated as described by Rod Vance in the other answer. Similar to the pi0 having mass but decaying to two photons with an angle between them to balance energy and momentum, which is what mass is in special relativity.

Is photon truly a photon only in vacuum?

A photon is always in a "vacuum" unless it is interacting. If the interaction is elastic it retains its energy and momentum .
A: To answer your title, "Is photon truly a photon only in vacuum?", yes indeed this is true. In a medium, light is a quantum superposition of pure photons and excited matter states. 
You are absolutely right that only particles with nonzero rest mass are observed at relative speeds less than $c$. A stronger statement ican be made: a particle's rest mass is nought if and only if it is observed travelling at a relative speed of $c$ and it is massive if and only if the observed relative speed is less than $c$.
Light in a medium has a rest frame - i.e. one travelling at speed $c/n$ relative to the medium, where $n$ is the medium's refractive index. So, one can calculate the light's rest mass by working its energy in this frame out. I show how to do this in my answer here and an optical light quasiparticle in a typical medium $n=1/5$ has a rest mass of about $m_0=3\times 10^{-36}{\rm kg}$ or about 4 millionths of an electron mass.
