What is the correct definition of the Gibbs free energy of activation?
There are a lot of possible definitions and it possibly depends very much on the case and the context you are looking at. However, there are the definitions provided by the IUPAC goldbook, which can be considered the official ones.

The first definition you suggested is correct so far and known as (standard) Gibbs energy of activation (standard free energy of activation) $\Delta^\ddagger{}G^\circ{}$:


  The standard Gibbs energy difference between the transition state of a reaction (either an elementary reaction or a stepwise reaction) and the ground state of the reactants. It is calculated from the experimental rate constant  $k$ via the conventional form of the absolute rate equation:
  $$\Delta^\ddagger{}G^\circ{} = \mathcal{R} T \left[\ln\left(\frac{\mathcal{k}_\mathrm{B}}{h}\right) − \ln \left(\frac{k}{T} \right)\right]$$
  where  $\mathcal{k}_\mathrm{B}$ is the Boltzmann constant and  $h$ the Planck constant ($\frac{\mathcal{k}_\mathrm{B}}{h} = 2.08358\cdot 10^{10}~\mathrm{K^{−1}s^{−1}}$). The values of the rate constants, and hence Gibbs energies of activation, depend upon the choice of concentration units (or of the thermodynamic standard state).


Your second definition refers to the definition of enthalpy of activation, $\Delta^\ddagger{}H^\circ{}$:


  The standard enthalpy of activation $\Delta^\ddagger{}H^\circ{}$ is the enthalpy change that appears in the thermodynamic form of the rate equation obtained from conventional transition state theory. This equation is only correct for a first order reaction, for which the rate constant has the dimension reciprocal time. For a second order reaction, for which the rate constant has the dimension (reciprocal time) × (reciprocal concentration), the left hand side should be read as  $k c^\circ$, where $c^\circ$ denotes the standard concentration (usually  $\mathrm{1 mol\cdot dm^{−3}}$).
  $$k = \frac{\mathcal{k}_\mathrm{B} T}{h} \exp\left\{\frac{\Delta^\ddagger{}S^\circ}{\mathcal{R}}\right\} \cdot\exp\left\{\frac{-\Delta^\ddagger{}H^\circ}{\mathcal{R}T}\right\}$$
  The quantity  $\Delta^\ddagger{}S^\circ$ is the standard entropy of activation, and care must be taken with standard states. In this equation  $\mathcal{k}_\mathrm{B}$ is the Boltzmann constant,  $T$ the absolute temperature,  $h$ the Planck constant, and $\mathcal{R}$ the gas constant. The enthalpy of activation is approximately equal to the activation energy; the conversion of one into the other depends on the molecularity. The enthalpy of activation is always the standard quantity, although the word standard and the superscript $^\circ$ on the symbol are often omitted. The symbol is frequently (but incorrectly) written  $\Delta{}H^\ddagger{}$, where the standard symbol is omitted and the  $\ddagger$ is placed after the  $H$.


The activation energy (Arrhenius activation energy) $E_a$ itself is defined as:


  An empirical parameter characterizing the exponential temperature dependence of the rate coefficient, $k$,  $$E_a = \mathcal{R} T^2 \frac{\mathrm{d}(\ln k)}{\mathrm{d} T}$$, where $\mathcal{R}$ is the gas constant and $T$ the thermodynamic temperature. The term is also used for threshold energies in electronic potential surfaces, in which case the term requires careful definition.


And while we are at it, entropy of activation $\Delta^\ddagger{}S^\circ$:


  The entropy change that appears in the thermodynamic form of the rate equation obtained from conventional transition-state theory.




Additional note and general warning
Although there are the official definitions, the term activation energy is often used as a synonym for many kinds of different barriers. One has to be very careful when reading articles, which definition is used.
In computational chemistry articles you may also find the terms electronic energy of activation, zero-point corrected energy of activation, solvent corrected energy of activation, ..., etc., pp., and many more.