I did some testing that seems to confirm my suspicion that there is no difference at all between the behavior of these UI elements.
First, I created a "Test Account" user and added it to the organization. This new user sees the single existing organization board, listed under "Other Org-Visible Boards."
As the admin user for the organization, I create two new boards using the two UI elements like so:
Note the yellow icon and message, "This board will be org visible," in both instances. Clicking "change" shows two other options for a new board:
All three visibility options are available and the default option is "org visible"* regardless of which of the two buttons is used to create the board. After creating the new boards, while logged in as the admin user, refreshing the organization page shows all three boards listed under "My Org Boards" despite having used the button under "Other Org-Visible Boards" to create Test Board 2:
When logged in as "Test Account," on the other hand, all three boards are initially listed under "Other Org-Visible Boards:"
After joining a board, it moves from "Other Org-Visible Boards" to "My Org Boards" as expected; e.g., joining Test Board 2 causes it to move to "My Org Boards." Out of curiosity, I repeated the process with the Test Account user creating the boards, and observed the exact same behavior; the user who creates the board is added as the initial board administrator and no other users are initially added to the board.
This leads me to conclude that organization boards created using either link behave in the exact same way.