Is the difference between heliocentric and geocentric velocities a matter of the coordinate system? Or are these kinds of velocities different? For example, the speed of a comet at perihelion is 71km/s. Is this speed geocentric or heliocentric?
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Yes, the difference is exactly that. The precise meaning here cannot be determined with certainty, it is a matter of convention.
With meteors (and their parents bodies in this context) true geocentric speed $v_g$ is most commonly reported. Meteors can only be detected in the Earth's atmosphere, and this is the speed measured from ground-based observations. It might or might not be corrected for the Earth's rotation ($\approx 460 \text{ m/s} \cdot \cos \phi$ where $\phi$ is the geographical latitude, a relatively small but measurable effect).
It is also common to calculate and report the "geocentric speed at infinity", that is, outside the Earth's gravity well but otherwise at the same location. From conservation of mechanical energy that is $v_\infty = \sqrt{v_g^2 - v_0^2}$, where $v_0 \approx 11.2 \text{ km/s}$ is the escape speed from the surface.
When talking about perihelions or orbits of comets, the obvious choice is the heliocentric or barycentric coordinate system. Geocentric speeds far from the Earth can be computed as well, but are not very useful. I would interpret the sentence "the speed of a comet at perihelion is $71 \text{ km/s}$" as a heliocentric / barycentric speed, but I cannot comment on whether this is what the author actually meant.
Yes, the difference is a difference of coordinate systems. The Earth moves with respect to the sun, and velocities depend on the choice of coordinate system.
In your particular case, it seems likely that 71 km/s is a geocentric velocity as it is the maximum geocentric velocity that a body could reach at one AU from the sun, if it falls from rest.
But, of course, a comet that has a perihelion closer to the sun could have a heliocentric velocity that is 71km/s.
Neither the geocentric, nor the heliocentric coordinate systems define an inertial frame of reference. This means that in these frames of reference, Newton's first law doesn't work. But the sun is close to the centre of mass of the solar system, which is very nearly an inertial frame.
