Do we have relativistic mass even when resting on Earth?

I think your confusion stems from the fact that many old textbooks use the term "relativistic mass" misleadingly. In the realm of special relativity there is in fact only one mass (rest mass), which is an inherent property of any particle/object, defined over the energy-momentum relation:

$ m = \frac{\sqrt{E^2 - (pc)^2}}{c^2} $

This mass is invariant under a Lorentz transformation, as the energy-momentum relation is invariant under a Lorentz transformation (if you're familiar with covariant notation: $p_{\mu} p^{\mu}$ is a scalar, otherwise just prove it explicitly for an arbitrary Lorentz boost and rotation (Lorentz boosts and rotations generate the proper Lorentz transformations)).

The misunderstanding comes from the fact that many authors mistakenly incorporate the $\gamma$ factor in the mass, thus obtaining a mass that depends on the reference frame. Find a great discussion in Relativistic mass (Wikipedia).

I hope this helps :)

Rest mass is an attribute of an object which remains the same in all frames. As you suggest, the total mass of an object is different from its rest mass if it has kinetic energy (meaning it is moving). However, in most cases where we measure a mass, we keep the mass holding still while we take the reading. In this case the object is motionless in the frame we are measuring in, so in that frame, its total mass and rest mass are identical.

If you were measuring the mass of a fast moving object (perhaps by measuring a momentum transfer), you would see the total mass as being higher than the rest mass because it had kinetic energy. However, in most cases we don't do this because it's harder to measure masses that are moving. Also, in realistic situations (sub-relatavistic speeds in the frame we are measuring in), the difference wouldn't matter for almost all meaningful measurements.

Supposing that you do want to use the convention where "relativistic mass" is a valid concept (because I don't think that "don't do that" suffices as a complete answer to your question):

You ask, does an object have a relativistic mass when it's sitting still on Earth? The answer is "yes, of course". An object always has a relativistic mass. For an observer at rest with respect to the object, though, its relativistic mass equals its rest mass. The speed with which the Earth zips around the Sun, or the galactic nucleus, or anything else doesn't matter; if you're conducting an experiment here on Earth, and the parts of your experiment aren't moving at ludicrous speed relative to one another, then you won't observe a relativistic mass that's noticeably different from the rest mass. That's what makes relativity relativity.