Which is the strongest base relative to ammonia?
Let's have a look at some of the definitions in the gold book and see how they relate to your problem.


  proton affinity
  The negative of the enthalpy change in the gas phase reaction (real or hypothetical) between a proton (more appropriately hydron) and the chemical species concerned, usually an electrically neutral species to give the conjugate acid of that species. Proton affinity is often, but unofficially, abbreviated as PA. 
  
  
  
  gas-phase basicity
  The negative of the Gibbs energy, $\Delta G_\mathrm{r}^\circ$ change associated with the reaction:
  $$\ce{B + H+ -&gt; B+-H}$$
  in the gas phase. Also called absolute or intrinsic basicity.


I was unable to find an exact definition for relative basicity, so I am going to assume based on the following reasoning. First we define the basicity for ammonia, $Bf(\ce{NH3})$, and an arbitrary amine $Bf(\ce{NR3})$. 
\begin{align}
\ce{NH3 + H+ &amp;~-&gt; NH4+}  -\Delta G_\mathrm{r}^\circ(\ce{NH4+}) &amp;
Bf(\ce{NH3}) &amp;= -\Delta G_\mathrm{r}^\circ(\ce{NH4+})\tag{1} \\
\ce{NR3 + H+ &amp;~-&gt; NR3H+}  -\Delta G_\mathrm{r}^\circ(\ce{NR3H+}) &amp;
Bf(\ce{NR3}) &amp;= -\Delta G_\mathrm{r}^\circ(\ce{NR3H+})\tag{2} \\
\end{align}

We can follow up and define the relative basicity of an arbitrary amine  to ammonia $Bf_\mathrm{rel}(\ce{NR3})$ as the difference of the independent basicities.
$$Bf_\mathrm{rel}(\ce{NR3}) = Bf(\ce{NR3}) - Bf(\ce{NH3})\tag{3}$$

With (3) we would actually describe the following reaction:
$$\require{cancel}
\begin{align}
\ce{NR3 + \cancel{H^+} + NH4+ &amp;~-&gt; NR3H+ + \cancel{H^+} + NH3}\\
\ce{NR3 + NH4+ &amp;~-&gt; NR3H+ + NH3}\\
\end{align}$$

So the relative basicity describes in this scenario how likely it is that the arbitrary amine is abstracting a proton from the conjugate base of ammonia, i.e. ammonium. The higher the relative basicity, the stronger the base.
The conjugate acid $\ce{NR3H+}$ is the strongest if its parent compound has the smallest basicity, meaning most likely to loose the proton. The weakest conjugate acid is obviously the strongest base.

But what have we actually compared? In principle we have discussed with this comparison how likely it is for the amines to bind a proton and we have chosen to do that in comparison with ammonia. We could chose to do the comparison to water and with a couple of transformations and simplifications we would arrive at a $\mathrm{p}K_\mathrm{a}$ which we would then compare. To make things easier, we could also directly compare the values of $\Delta_\mathrm{r}^\circ$ in (1) and (2); the lower the value, the higher the basicity.

If we now choose to compare the enthalpies instead of the Gibbs energies, then we would be ultimately comparing (gas-phase) proton affinities. We would be still looking at the same reactions, just a different characteristic property. We would still be observing the same trend.

TL;DR It does not matter if you look at proton affinities or relative basicity, you are still comparing the same effect with different numbers, that are all linked.