To answer your title question, the first "programmable, digital computer" was designed by Charles Babbage and Ada Lovelace, known as the Analytical Engine. Turing, von Neumann, and other computer scientists and mathematicians approached the concept of a programmable computer both independently and building on the work of Babbage, Lovelace and their contemporaries. Turing devised the concept of a general-purpose computer in the 1930s, which the Analytical Engine was theorized to be. A quote exists from the second link which illustrates how the modern computer was theorized through Babbage's engine:
"Charles Babbage started work on his Analytical Engine in the mid-1830s, with the idea of creating a new calculating machine that could “eat its own tail”, by which he meant that it could modify its calculation while it was running. It would do this by pausing during a calculation, and using the values it had already determined to choose between two possible next steps. Babbage listed the basic operations that such a machine, with a large enough memory, would need if it were to execute “the whole of the developments and operations of analysis”, in other words any calculation that could be conceived of at the time. We now know that the basic operations that he described are what are needed to compute anything that can be calculated by any modern computer. This means that the Analytical Engine would have been, in modern terms, a general-purpose computer, a concept first identified by Alan Turing in the 1930s."
Many mathematicians and computer scientists interpreted this concept through their distinct vision, and they also built on it through various manifestations.
Hope this helps!