There are a least a couple problems with your construction.
First, your construction uses $U$ and the objects $a$, $b$ and $z$. Where do they come from? If $U$ is still a functor and $a$, $b$ and $z$ are objects of $\mathcal{D}$, then we're back where we started with regards to the original definition. You may just be treating $U(a)$ and the rest as formal symbols, which is fine, but that means that this has nothing to do with the original construction.
Second, you don't have a complete list of the objects of the arrow category of $\mathcal{C}$. You said we morphisms $U(g) : U(a) \to U(b)$ and $U(h) : U(a) \to U(z)$, which means there are (at least) more objects in the arrow category. Also, you've neglected to mention the four identity arrows for $x$, $U(a)$, $U(b)$ and $U(z)$.
Moreover, there are a few compositions missing, though you may be implicitly saying that $U(g) \circ f_3 = f_2$ and $U(h) \circ f_3 = f_1$. We have to assume that this is true for $U(g)$ (really $\langle id_x, U(g) \rangle$) and $U(h)$ to be morphisms in the arrow category.
Now including the missing objects, $\langle x, f_3, U(a) \rangle$ is no longer initial: there's no morphism from $\langle x, f_3, U(a) \rangle$ to $\langle x, id_x, x \rangle$ since there's no morphism $U(a) \to x$ in $\mathcal{C}$. Instead, $\langle x, id_x, x \rangle$ is initial (assuming the necessary diagrams commute).
The answer to your question, though, is yes. Universal properties can be defined using a single category (no arrow category needed). In fact, initial objects encompass all or almost all universal properties. It's all just a matter of changing which category you want the initial object to be in. The construction with the functor $U$, however, is simply a useful special case (for the category $(X \downarrow U)$) that applies to many situations. For example, limits can be defined using that construction.
It's just helpful to work out the details for special cases so you get a good general idea of what's possible. "Initial objects" be themselves don't seem very interesting, but using some special categories derived from other data (in this case, the functor $U: \mathcal{D} \to \mathcal{C}$), you get something much more interesting.
The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.
Best Answer
What you are looking for are local objects. I have once searched for reference about them and quickly gave up as I have found nothing usable, only the nLab page. Therefore I will rather provide the insight and examples you have also asked for:
Let $\mathcal{C}$ be a category and $J\subset\operatorname{Ar}\mathcal{C}$ be a class of morphisms, then:
Definition ($J$-local object): An object $c\in\operatorname{Ob}\mathcal{C}$, for which $\operatorname{Hom}_\mathcal{C}(-,c)\colon\mathcal{C}^\mathrm{op}\rightarrow\mathbf{Set}$ maps all morphisms in $J$ to isomorphisms (bijections), is a $J$-local object.
Definition ($J$-local morphism): A morphism $f\in\operatorname{Ar}\mathcal{C}$, for which $\operatorname{Hom}_\mathcal{C}(f,c)$ is an isomorphism (bijection) for all $J$-local objects $c\in\operatorname{Ob}\mathcal{C}$, is a $J$-local morphism.
To give a few results: Immediatly from the definitions, we have:
Corollary: Every morphism in $J$ is a $J$-local morphism.
Corollary: Every isomorphism is a $J$-local morphism.
Since limits preserve isomorphisms, we have:
Lemma: Limits preserve $J$-local objects and colimits preserve $J$-local morphisms.
Since bijections fulfill the two-out-of-three property, we have:
Lemma: $J$-local morphism fulfill the two-out-of-three property.
Corollary: Together with the class of $J$-local morphisms, $\mathcal{C}$ is a category with weak equivalences.