I checked and yes I am (in my personal twitter).
I am creating this map for an organization. Do I need to be logged in into the Twitter account of the organization?
(I was also wondering if it is possible to create some kind of story map in the twitter map as well? Because I have pictures and locations of sites which I need show on the map but the web map also needs to have the functionality where people take pictures of sites and can upload them onto the web map using twitter)
I created an application following the steps from the blog you listed. I did this in the FireFox browser. When I got to this step:
I clicked on Save and then on View. In my screen afterwards, there is a place for me to sign in to Twitter:
Do you see this part? After I click on the blue link, I can sign into Twitter.