The assumption is strictly stronger than Robinson arithmetic. It allows us to assume that Π01 sentences from the arithmetic hierarchy have a "privileged ontological status". Because of the way I formulated the question, Π02 (unintentionally) fails to achieve a similar status. I conclude this because of Harvey Friedman's manuscript on "Order Invariant Relations and Incompleteness":
DEFINITION 4.4. A Π01 sentence is a sentence asserting that some given Turing machine never halts at the empty input tape. A Π02 sentence is a sentence asserting that some given Turing machine halts at every input tape.
Added 16. April 2016: It turns out that the "privileged ontological status" of Π01 sentences (or rather the absence of this status for Π02 sentences) is more than just an arbitrary unintentional artifact of formulation details. A Turing machine has a finite number of internal states, and an infinite tape, but it is unable to decide whether an infinite input is really infinite, or whether it would turn out to be finite after all, if it only continued reading it long enough. Here is another explanation:
I was recently surprised to find out that one can consistently accept the existence of Turing machines, and reject the existence of Turing machines with access to an oracle for deciding the halting problem problem of a Turing machine. This is because the oracle can lie (but one cannot prove that it lies) and claim that a non-halting computation would actually halt, and then taking forever while answering with an infinite number, when asked for a bound on the number of steps. (I realized this after writing a technical justification for the excuse: Then I might justify my doubts about Π02 sentences by explaining that I'm unsure about how to separate finite inputs from infinite inputs, and hence am unsure whether quantification over the inputs is well defined. That excuse itself arose from a conversation with another Thomas.)
If one is willing to grant the existence of a succession of increasingly more powerful oracle Turing machines, then one seems to be able to get ontological commitments "arbitrary close to the Church-Kleene ordinal", as explained here:
To answer your question, I claim that statements about Turing machines with halting oracles have definite truth values, just as “clearly” as statements about ordinary TMs do. The argument is simply this: you agree that any given Turing machine, on any given input, either halts or doesn’t halt? Well then, every call to a halting oracle has a definite right answer. Therefore, the operation of a TM with a halting oracle is mathematically well-defined at every point in time. Therefore, by the same intuition you used for ordinary TMs, you should conclude that the oracle TM either halts or doesn’t halt, and that there’s a definite fact of the matter about which it does.
Continuing recursively in this way gets you to definite truth-values for all Π0k sentences, for any value of k—and actually further, to Π0α sentences, where α is any computably-definable ordinal. But it doesn’t get you beyond that: it stops at the supremum of the computable ordinals, which is called the Church-Kleene ordinal. Beyond that lies the enchanted realm of the Axiom of Choice and the Continuum Hypothesis, for which I don’t think one has to admit the existence of any well-defined truth value (though neither is one prohibited from doing so).
Edit: On closer reading, this is actually a misinterpretation from my side. We don't get the well-order of those ordinals (we have to somehow know that they are well-ordered), we just get that Π0α sentences have definite truth values (for the mentioned ordinals α).
These two answers are not necessarily in the direction that I initially intended this question, but they seem to describe the ontological commitments that people in practice derive based on assumptions related to the existence of (powerful oracle) Turing machines. This answer is quite unpolished, but who cares anyway? Later...