"...the term ' telegraph' means a wire or wires used for the purpose of telegraphic communication, with any casing, coating, tube, or pipe inclosing the same, and any apparatus connected therewith for the purpose of telegraphic communication."
In a legal sense above the telegraph is just a bunch of wires for sending messages. (It's sort of like the internet being a system of tubes.) But already the message is being conflated with the wiring instead of the technology or the data. This is where the other senses of wire start to emerge. So it appears that first the message becomes "a wire" then, the act of sending that message becomes the act "to wire."But it's not that clean cut. The only hard line here is the invention of the telegraph in 1809 before which no messages were sent over wires of any kind.
The book The Progressive Dictionary of the English Language by Samuel Fallows printed the following definitions in 1885:
Wire (wir): To send by telegraph, as a message ; to telegraph ; as in wire a reply. To communicate by means of the telegraph; to telegraph ;
Wire (wir), f. i. To communicate by means of the telegraph;
But common usage in print was erratic. In 1891 J. Bucknall Smith wrote the book Wire, it's Manufacture And Uses. It was a canonical book on the production of and the use of what he still called "wire rope" even when describing braided electrical wire. Chapter IV "Electrical Conductors" is entirely about it's use for electrical an telegraph applications. Throughout the book wire is used only as a noun. Even when the wire was installed the wire was "drawn" the building was not "wired". In the 1898 book A Dictionary of Electrical Words, Terms and Phrases by Edwin James Houston it still appears only as a noun as it often still did in periodicals. I assume therefore that use as a verb was still uncommon until the early 1900s.
1874 - Message of the Governor of Virginia to the General Assembly (1/1/1874)
"At once I wired his excellency, the governor of West Virginia, for permission to transport troops through that State..."
1877 -Arctic Expeditions from British and Foreign Shores, David Murray Smith
"Lieutenant Hamilton "wired " a message to the tender, inquiring whether Hartnell had come on board."
1882 - Chambers Journals, August
"But a day or two actually going by, and she hearing no more, she wired a message of inquiry to the offices."
1885 - The Mountain Kingdom, David Lawson Johnstone
"At the first telegraph-station I wired a message to my mother, telling her of our safety."
1888 - Votes & Proceedings New South Wales Parliament (8/23/1888)
"Instructions were sent by wire on 20th instant"
1892 - Proceedings of the Brotherhood of Locomotive Firemen and Enginemen
" Mr. Clark immediately wired a message to Trinidad : " Return to work. Send a committee to my office at Omaha and I will settle your grievances."
1894 - A Born Soldier, John Strange Winter
"I will send a wire to Winthrop now. Perhaps you will send it off for me."
1894 - The Greek Madonna, C. W. de Lyon Nichols
"Madeline could not have wired a message to a more consenting partner..."
1894 - The Maiden's Progress, Violet Hunt
"Portrait painter, you know! He is in the office. Shall I send the wire for you?"
1895 - Pall Mall Magazine, February
"...yet in the morning, after reconsidering the question the author actually did send the wire to his brother instead."
1897 - McBride's Magazine, August
"Very good Sir. I will send the wire off at once."
1899 - The Lady's Realm, July
"Griffiths received the news calmly, gave his orders in a curt, confident way, then hurried down to the railway station to send the wire to Himla"