After thinking, and reading other references and re-reading the papers I mentioned, I have found a sufficient explanation (at least to my care):  Both instantons/monopoles are solutions to corresponding equations of motions from associated actions, and they "bloom" from an overarching SUSY action.

Witten formulated *"twisted N=2 Supersymmetric Yang-Mills"*, a TQFT with SUSY (supersymmetry), which leads to the Donaldson invariants. This used an $SU(2)$-bundle over $X$ along with a gauge field (connection $\omega$) and matter fields (bosonic $\phi,\lambda$ and fermionic $\eta,\psi,\zeta$), and gave the *Donaldson-Witten action functional* $S_{DW}=\int_Xtr(\mathcal{L})$,  
$\mathcal{L}=\frac{1}{4}F_\omega\wedge(\ast F_\omega+F_\omega)-\frac{1}{2}\zeta\wedge[\zeta,\phi]+id^\omega\psi\wedge\zeta-2i[\psi,\ast\psi]\lambda+i\phi d^\omega{\ast d^\omega}\lambda-\psi\wedge\ast d^\omega\eta$.  
This has associated partition function $Z_{DW}=\int e^{-S_{DW}/g^2}D\Phi$ (here $\Phi$ denotes the space of aforementioned fields), where $g$ is a coupling constant that is the key here for answering our question.  The "blooming" of this action functional is beyond the scope of my intentions and probably of MathOverflow, so I won't question it.

In weak coupling ($g\rightarrow 0$, known to physicists as the ultraviolet region), the action localizes to the classical Yang-Mills $S_{YM}=\int_Xtr(F_\omega\wedge\ast F_\omega)$ and have the equations of motion $F_\omega=\pm\ast F_\omega$.  The solutions *are* the Donaldson *instantons*.

Now apparently, when we instead look at strong-coupling ($g\rightarrow\infty$, known to physicists as the infrared-region), the Seiberg-Witten equations should arise (a "duality" in Witten's TQFT). Indeed, Seiberg and Witten showed that this infrared limit of the above theory is equivalent to a weakly-coupled $U(1)$-gauge theory (the $SU(2)$-gauge group is spontaneously broken down to the maximal torus). **Perhaps here is where a better understanding would be desirable** (buzzwords 'asymptotic freedom' and 'symmetry breaking' appear).  
Anyway, some physics-technique stuff happens (the previous paragraph can be described as "condensation of monopoles"), and we must consider a spin-c structure (which all of our oriented 4-manifolds have, whereas a spin structure would not allow us to consider all 4-manifolds); note that $Spin^c(4)=(SU(2)\times SU(2))\times_{\mathbb{Z}_2} U(1)$. This gives the data: $U(1)$-gauge field $A$ and positive spinor field $\psi$ (as written in the original post). The pair $(A,\psi)$ is a *monopole* when it minimizes an action $S_{SW}$, i.e. are time-independent solutions to equations of motions (the Seiberg-Witten equations).  The action here is $S_{SW}=\int_X(|d^A\psi|^2+|F_A^+|^2+\frac{R}{4}|\psi|^2+\frac{1}{8}|\psi|^4)$, with scalar curvature $R$.

I hope this post is not as confusing as I think it can be.