How do US electioneering prohibitions apply to digital messages?
A comprehensive answer for the US would require reviewing each state's law, but since you included Ohio in the question, it doesn't.
Section 3501.35 prohibits loitering or congregating near polling places. It goes into more detail on specific activities by stating "During an election and the counting of the ballots, no person shall do any of the following:"
(1) Loiter, congregate, or engage in any kind of election campaigning within the area between the polling place and the small flags of the United States placed on the thoroughfares and walkways leading to the polling place, and if the line of electors waiting to vote extends beyond those small flags, within ten feet of any elector in that line;
(2) In any manner hinder or delay an elector in reaching or leaving the place fixed for casting the elector's ballot;
(3) Give, tender, or exhibit any ballot or ticket to any person other than the elector's own ballot to the precinct election officials within the area between the polling place and the small flags of the United States placed on the thoroughfares and walkways leading to the polling place, and if the line of electors waiting to vote extends beyond those small flags, within ten feet of any elector in that line;
(4) Exhibit any ticket or ballot which the elector intends to cast;
These first 4 activities all require physical presence at a polling location to be in violation and therefore clearly do not apply to a remotely sent text message. The fifth element listed could be argued to apply when you consider the law says "no person shall do any of the following:"
(5) Solicit or in any manner attempt to influence any elector in casting the elector's vote.
However, this 5th element must be considered within the context of the others: Clearly the intent of the law is to prohibit voter intimidation and influence via active electioneering at the polling place. Passive receipt of a message on an electronic device in your pocket has far less effect than a sign wielding campaigner chanting in your face from 11 feet away. (which surprisingly would apparently be legal outside the flags?!)
The first "attempt to influence" might be nothing more than a buzz in your pocket that you can easily ignore, but the other could be a very intimidating experience.
Truly, if the intent is to block anything remotely influential you'd have to prohibit all political advertising during any time someone might be in line for voting for fear they could view it on a portable electronic device. A reductio ad absurdum example for states with mail in ballots would mean you'd have extend that ban back to the time mail in ballots become available. (Which I actually don't oppose!)
Finally, to make a case of electioneering against some person or campaign office you'd have to prove intent to specifically target people in line at a polling place. A widely broadcast (and easily ignored) spam message such as "Vote for Bob" would lack the mens rea to be prosecuted under the Ohio state law as it is currently written.
For more information on the history and purpose of electioneering laws, please read this article.