13x8 rims and tire question

Will any 13 tire fit 13x8 rims
I have a set of Hoosier td 23/7.0-13
And a set hossiser Comanche 22.5/6.0-13
They fit and air up but will it hold?
New to the mini racing any advice would help
Track calls for Hoosier 790 but like to use these for lap time
Just want to be safe and learn different in tires and rim different
Do I have 13x8? How do you measure the correctly
I laid they on a flat surface and measured with lip to lip?
Or is it the inside tire area to measure ?
Thanks in advance