OP: "if the first cause is by definition uncaused, what possible principle could serve to differentiate between these two kinds of first causes?"
This can be explained from a phenomenological point of view. Beings—as things that are—objectively exist as "the copula of a judgement", by cognition joining concepts with predicates, i.e. from Kant's Critique of Pure Reason A598/B626, original here:
Being is evidently not a real predicate, that is, a conception of
something which is added to the conception of some other thing. It is
merely the positing of a thing, or of certain determinations in it.
Logically, it is merely the copula of a judgement.
So, for example, humans make a judgement on the moon and its existence begins, notwithstanding its obvious age.
On the other hand, the judge, a subjective being as in Dasein, is of a higher order of existence, temporal and conscious.
Let us suppose these beings come from Being. In line with the principle of non-contradiction, for beings to come from being, being must not have the characteristics of beings. If Being is the foundation of beings it cannot itself be like a being, otherwise the thing founded would be in its own foundation.
Being, as the basic theme of philosophy, is no class or genus of
entities; yet it pertains to every entity. Its 'universality' is to be
sought higher up. Being and the structure of Being lie beyond every
entity and every possible character which an entity may possess.
Being is the transcendens pure and simple. (Heidegger, Being & Time, H.38)
So human Dasein-existence and consciousness, the "I am", emerges from something called Being, which is distinctly different from a being. E.g. a dynamic process of emergence rather than a static concept, although Heidegger is more philosophically idealist and does not admit any determination of Being. From Pathmarks:-
[p.97] The nothing is the "not" of beings, and is thus being,
experienced from the perspective of beings.
[p.253] But the clearing itself is being.
The clearing is the noumenal abstraction from which Dasein—amid the twofold concealment of refusal or obstruction—discerns its objects, correctly or mistakenly.
The principle the OP quotes "that everything that begins to exist has a cause" is the principle of sufficient reason. Every being is predicated as having a reason, but Being itself is not a being and is therefore exempted from the principle. Derrida refers to Being as transcendental signified.
Since reason is part of making things intelligible to consciousness reason has some consonance and involvement in Being. So rather than there being a reason for being, rather reason is itself being-like.
The following quotes from Heidegger's The Principal of Reason (1957) sketch out the progression from the principle of reason to its alignment with Being.
Being itself is left as a mystery. More can be said about what it is not than what it is, not that it actually is in the normal sense. It is deduced from "beings come from Being" and Being must be completely different to a being. It is a conclusion of philosophical rigour to which I would be interested to hear objections.
[p.11] In its short formulation [Leibniz's 1686] principle of
reason reads:
Nihil est sine ratione; nothing is without reason. ... Therefore, according to what the principle itself tells us, it is the sort of
thing that must have a reason. What is the reason for the principle of
reason?
[p.12] But what are we getting ourselves into if we take the principle
of reason at its word and move towards the reason of reasons? Does not
the reason of reasons press forward beyond itself to the reason of
reason of reasons? If we persist in this sort of questioning, where
can we find a respite and a perspective on reason? If thinking takes
this path to reason, then surely it can't help but fall intractably
into groundlessness.
[p.44] "Nihil est sine ratione": "Nothing is without reason."
Every being has a reason. The subject of the principle of reason is
not reason, rather: "Every being"; this is predicated as having a
reason. The principle of reason is, according to the ordinary way of
understanding it, not a statement about reason, but about beings,
insofar as there are beings.
[p.49] "Nothing," that is, no being whatsoever "is—without
reason."
[p.50–51] [Finally] we hear the principle of reason in a different tonality.
Instead of "Nothing is without reason," it now sounds like this:
"Nothing is without reason." The pitch has shifted from the
"nothing" to the "is" and from the "without" to the "reason". The word
"is" in one fashion or another invariably names being. This shift in
pitch lets us hear an accord between being and reason. Heard in the
new tonality, the principle of reason says that to being there belongs
something like ground/reason. The principal now speaks of being. What
the principal now says, however, easily falls pray to a
misinterpretation. "Ground/reason belongs to being"—one might be
inclined to understand this in the sense of "being has a reason," that
is, "being is grounded." The popularly understood and presumably valid
principium rationis never speaks of this. According to the principal of reason, only beings are ever grounded. On the contrary,
"ground/reason belongs to being" is tantamount to saying: being qua
being grounds. Consequently, only beings ever have their grounds.
The new tonality reveals the principle of reason as a principle of
being. Correspondingly, if we now discuss the principle in the new
tonality, we move in the realm of what one can, with a general term,
call the "question of being."