I recreated the Binary Systems logo using a combination of Photoshop and Illustrator, but mostly Illustrator. The only thing I used Photoshop for was to create the image of the binary stars, which I then brought into Illustrator. I found an image of a lens flare at https://www.brusheezy.com/textures/46510-lens-flare-textures-by-brandondorf and used that as the basis for making the stars. I added the horizontal and vertical streaks by hand. Due to my pretending to be an artist, getting those streaks to look right took me much longer than I expected.
After I created the binary stars in Photoshop, I brought them into Illustrator where I created the rest of the Binary Systems logo by using the Space Age font and creating the red line. I eyeballed the placement of everything in order to recreate the original logo as faithfully as possible. Here is the result, and the original logo to compare with. What do you think?
Now would be a good time to also mention that I put all of my artwork into the Starflight GitHub Project, for you to do whatever you want with them. For example, the files for this logo can be found inside the Illustrator\Binary Systems folder.