Legally speaking a printout of your I-94 satisfies the letter of the law. Most people do not carry this around, however. The only people who will enforce this law are federal immigration agents (Border Patrol -- USBP -- or Immigration and Customs Enforcement -- ICE). You will probably never see either of these unless you are near the border with Canada or Mexico, where you may encounter USBP.
as I understand I need some photo ID on me in any case
Not really. For state and local police, your only requirement is to identify yourself, and only under certain circumstances, such as being suspected of a crime. You are not required to have any particular document to identify yourself. Details vary from state to state.
Having your Hungarian national ID can't hurt, but many people may be too unfamiliar with it to accept it. They may be more likely to accept whatever identification you receive from your J-1 sponsor. If someone doesn't know what to make of it you can help them out a bit by saying "it's like a passport card."
Another thing to think about is proof of age for buying alcohol. The specific acceptable documents vary from state to state, but I suspect that the national ID would not be formally acceptable in most places.
it's unclear what documents this actually refers to, and whether they should be original or if copies or electronic versions (on smartphone) are acceptable.
This is clarified in the Code of Federal Regulations (8 CFR 264.1) (with some omissions):
(b) Evidence of registration. The following forms constitute evidence of registration:
Form No. and Class
I–94, Arrival-Departure Record—Aliens admitted as nonimmigrants; aliens paroled into the United States under section 212(d)(5) of the Immigration and Nationality Act; aliens whose claimed entry prior to July 1, 1924, cannot be verified, they having satisfactorily established residence in the United States since prior to July 1, 1924; and aliens granted permission to depart without the institution of deportation proceedings.
Note to paragraph (B):
In addition to the forms noted in this paragraph (b), a valid, unexpired nonimmigrant DHS admission or parole stamp in a foreign passport constitutes evidence of registration.
But the I-94 is no longer physically given to most people. 8 CFR 1.4:
Definition of Form I–94
The term Form I–94, as used in this chapter I, includes the collection of arrival/departure and admission or parole information by DHS, whether in paper or electronic format, which is made available to the person about whom the information has been collected, as may be prescribed by DHS. The following terms, when used in the context of the Form I–94, are clarified as to their meaning to accommodate the collection of such information in an electronic format.
(a) The terms “annotate,” “note,” “indicate on,” “stamp,” and “endorse,” unless used in part 231 of this chapter, include, but are not limited, to DHS amending, including or completing information in its electronic record of admission, or arrival/departure. For purposes of part 231, the term “endorse” includes but is not limited to the submission of electronic departure data to CBP.
(b) The terms “completed,” “completely executed” and “completed and signed” include, but are not limited to, DHS completing its collection of information into its electronic record of admission, or arrival/departure.
(c) The terms “issuance” and “given” include, but are not limited to, the creation of an electronic record of admission, or arrival/departure by DHS following an inspection performed by an immigration officer.
(d) The term “original I–94” includes, but is not limited to, any printout or electronic transmission of information from DHS systems containing the electronic record of admission or arrival/departure.
(e) The terms “present,” “presentation,” or “submission” of a Form I–94, unless they are used in § 231.1 or § 231.2 of this chapter, include, but are not limited to, providing a printout of information from DHS systems containing an electronic record of admission or arrival/departure. For purposes of § 231.1 of this chapter, the terms “present” or “submission” of the Form I–94 includes ensuring that each passenger presents him/herself to a CBP Officer for inspection at a U.S. port-of-entry. For the purposes of § 231.2 of this chapter, the terms “present,” “submit,” or “submission” of the Form I–94 includes ensuring that each passenger is available for inspection by a CBP Officer upon request.
(f) The term “possession” with respect to a Form I–94 includes, but is not limited to, obtaining a copy or printout of the record of an electronic evidence of admission or arrival/departure from the appropriate CBP systems.
(g) The terms “surrendering,” “turning in a Form I–94,” and “departure I–94” includes, but is not limited to, complying with any departure controls under 8 CFR part 215 that may be prescribed by CBP in addition to the submission of electronic departure data to CBP by a carrier.
Unfortunately, many federal immigration officers misunderstand or at least misrepresent the requirements of the law as a requirement to show a full set of documents justifying your present immigration status, i.e., including your DS-2019, even though that is not what the law actually says. See for example the experience of Mohanad Elshieky.
You are exceedingly unlikely to encounter these people, unless (as already noted) you are near the border. If you do, and they write you a ticket despite your having a printout of the I-94, you ought to be able to fight the ticket successfully, because that is all the law requires you to have.