Texas and Ohio are both take part in the Drivers License Compact. This agreement requires members to report back to the state in which a motorist is license in any out of state traffic violations. Thus the TX DPS or courts should report to the OH Bureau of Motor Vehicles (BMV) this speeding ticket.
We have read that Ohio does not assign points for out of state tickets, but will place the violation on your driving record. State laws can change or be amended though so I would recommend checking with the Ohio BMV to find out for certain how they handle out of state violations with regards to your driving record and points assessment.
As long as the car had insurance on it by its owner, which it apparently did, than it likely does not matter that you did not have insurance on yourself at the time. Your friend's insurance liability policy normally would cover you to drive the vehicle, unless there is an exclusion, as long as you had their permission.
Texas and Ohio do not report tickets directly to insurance companies. Traffic violation conviction information is placed on your driving record. Since in this case the speeding citation was to written to you, then it would be your driving record it would go on. If the citation had been for registration or a mechanical issue, it likely would have been written up to the vehicle and thus been placed on the owner's record.
If Ohio chooses to place this Texas speeding ticket on your driving record then when an insurance company next pulls your motor vehicle record (MVR) they will see it and depending upon the company's rating system your premiums could be affected.