@jlrdw @martinbean @snapey You all gave me great suggestions.
Hmmm ... the idea that an organization can do what it needs with its trainers without modifying the related user is a great idea.
And having a pivot organization to store all details specific to that organization-user relationship is also a great idea.
Ultimately this pivot table could be called collaborators, each collaborator could be a trainer or a student or an assistant, ... and have its own datas specitif to the related organization.
The idea to switch from one to another organization whilst logged in is also really great (like with Slack for example), but it would be interesting too that a trainer can for example see all his/her trainings on the same view : training A for organisation 1, training B et C for organisation 2, ... with the title, the dates, the number of students, ...
The trainers will necessarily have very little access to the application (only see their trainings) ... but if a trainer is also admin of his/her own training organization, he/her has to be able to manage his/her own organization's trainers.