What is the relationship between the two definitions of the difference between basicity and nucleophilicity?
The second set of definitions is the correct one.

Basicity is a thermodynamic property. In essence, the basicity is measured by determining the equilibrium constant of the following (generic) reaction:
$$\ce{B- + H2O &lt;=&gt; B-H + OH-}\tag{1}$$

The proton transfer that is the gist of this is an extremely fast process — faster than diffusion control due to the Grotthuss mechanism. You may have witnessed this during acid-base titrations: equilibrium is reached extremely rapidly. Another way to observe this is by adding carbonate bases to an acidic solution; $\ce{CO2}$ evolution is basically instantaneous. Therefore, there is no way to observe just one step (forward or reverse); you basically always have them both.

Nucleophilicity is a property that is related to the rapidness of the following, again generic, reaction:
$$\ce{Nu- + R-LG -&gt; Nu-R + LG-}\tag{2}$$

As similar as this may seem to the above, the lack of a reverse arrow may already show that this is tied to only one directiong; that it is not an equilibrium process but rather a kinetic one. Most of the time, there will be one species that is clearly a good nucleophile ($\ce{Nu-}$ in the scheme) and one that is only a good leaving group ($\ce{LG-}$ in the scheme). The reaction will be driven to completion and stay there.

To exemplify the difference, consider what would happen if you added hydroxide to the acid-base equilibrium $(1)$ versus additional leaving group to the nucleophilic substitution $(2)$. In $(1)$, the equilibrium would readjust; if the base is less basic than hydroxide it will be deprotonated, i.e. the reaction will be shifted back to the reactants. In $(2)$, typically nothing will happen, e.g. if you add $\ce{OPPh3}$ to a Mitsuobu reaction; showing the kinetic nature of the process.

Another way to look at the problem — and to separate the parameter basicity from the parameter nucleophilicity — is to look at the influence of steric effects. For example, consider ethanolate, propanolate and deprotonated isobutyl alcohol. In all three cases, the negative charge is unambiguously located on the oxygen atom. Thus, we would expect a thermodynamic property to be relatively equal for the threeof them. At the same time, there is a clear increase in steric hindrance from ethanolate to the isobutyl alcohol. This steric effect should manifest itself in reaction kinetics and lead to the latter being less nucleophilic than the former. Indeed, sterically hindered bases are often described as non-nucleophilic while the same structure with less hindrance often works fine as a nucleophile.

Finally, let’s work at debunking the claims laid out in the first set of definitions. First, basicity. After the Arrhenius and Brønsted-Lowry theories of acids and bases came the Lewis theory. In this, the reaction of a Lewis base with a Lewis acid — not necessarily a proton — is an acid-base reaction. One of many possible examples would be the formation of a transition metal complex such as $\ce{[Cu(NH3)4(H2O)2]^2+}$. Here, the ammine ligand is acting as a base, but it is donating to a non-hydrogen atom (namely copper). The claim that basicity only goes towards hydrogen is obviously false.

Unfortunately, it is not possible to fully debunk the second part since a nucleophilic attack on a proton is not really possible (that would indeed be the thermodynamic acid-base reaction). However, we may remember that most of the non-metals can act as the recipient of a nucleophilic attack; for example sulphur in a Swern oxidation, phosphorus in a Horner-Wadsworth-Emmons reaction or iodine in a Dess-Martin oxidation.