The ten day rule for the summer season does not apply unless there are more than four school members on the travel ball team that is coached by that school's coach (volunteer or paid, assistant or head, those factors don't matter). In assessing whether there are more than four school members on the travel ball team, one needs to consider any players who participated in school-sponsored middle school ball. And if there are more than four members from one school, any coaching contact by more than one school coach has to be aggregated when calculating the ten days.
Please note that if there is even one school player on the travel ball team, coaching the travel ball team for the outdoor fall season or for indoor ball is prohibited for HS coaches. Travel ball teams with more than four members from one school cannot participate in outdoor fall season (no matter who the coach is) but can participate in indoor ball as long as there is no HS coach from the school handling the indoor ball team.
I have heard that there is at least one Ohio HS coach who gets around the restrictions by resigning each year immediately upon completion of the HS season. The rumor is that the HS coach is then rehired by the School Board at a more convenient time of the year. I have no direct knowledge of that situation and I do not know what position OHSSA would take on that practice.