If the other toilets/ tubs/ sinks are draining fine, the exterior cleanout is going to be of little use. That's the cleanout to access between the house and the city line, and if it was clogged, other fixtures would be backing up.
You'll likely need to pull the toilet and run a snake/ auger down the line to remove the obstruction. The lead end of it needs to have a larger tip on it and rotate to remove the clog vs just punching a small hole into it.
Hopefully it's just a waste/ toilet paper clog and not something more serious, such as a root ball that's entered through a cracked drain pipe.
I use Wilson Plumbing for all of my repair and remodel work. 979.690.4219