I believe I'm getting closer to a better understanding of what I need.
Thanks to all of you for you help.
Here are the pieces of the puzzle:
*I have a blog/content on flippinfactory.wordpress.com
*I want to rebrand my blog. I no longer want the name/brand to be flippinfactory. I want to change it to curbsideconnections.
*I own the domain name curbsideconnections.com that I bought through godaddy, not wordpress.
*The domain name that I own (curbsideconnections) is just a parked page right now. There is no site or blog associated with it.
*I don't want to keep flippinfactory. In addition, it is set to expire in December.
If my understanding is clear, I have a couple of options to change from flippinfactory.wordpress.com to curbsideconnections.wordpress.com:
change the blog title and blog url of flippinfactory to curbsideconnections. Then, I can point (through the godaddy dashboard) the domain name that I own (curbsideconnections.com) to the wordpress blog. If I want to keep my subscribers and permalinks and redirect traffic, I will have to purchase the SITE REDIRECT. According to auxclass's answer above, visitors will see the new URL, not the old one.
follow all the steps above without purchasing the site redirect. If I don't purchase it then I'll lose my permalinks, subscribers, and traffic. I would be starting over. This option makes sense only if I don't have a new blog to export
I could create a brand new blog with curbsideconnections.wordpress.com and point the domain name that I own (through my godaddy dashboard, not through WP dashboard/support) to this new wp blog. I would keep the old blog (flippinfactory.wordpress.com) and create a featured sticky post that informs visitors of the change and tell them to go over to curbsideconnections. Subscribers and permalinks would not follow.
Option 4) (which seems to only apply if I had two blogs that I wanted to merge).
With this option, I could follow the directions for the MOVING BETWEEN WORDPRESS.COM BLOGS on the support page, which entails exporting the old blog and importing it into the new one. If done this way, I wouldn't keep followers or permalinks because I essentially created a new blog and moved the files of the old blog over to the new blog.
Given all the pieces of my puzzle, it still seems to me that SITE REDIRECT and DOMAIN MAPPING are the same thing. What, if any, is the difference between the two?