Can I create two Discovery forms (or of other type), one for older clients and one for newer clients, in a way that each of these clients see their appropriate version of the form?

Yes. Although you can only create one Discovery, Goals, Call Strategy, or Evaluation form as form type, you can create a second one of these forms as custom type forms (as many as you like, in fact). There is no important difference from the coach perspective between a Discovery type form and custom type forms, only a general classification.

In your example, you can leave the old Discovery form (as Discovery type) for your older clients and assign permission to the second custom Discovery form for your newer clients. Then assign the new Discovery form to a menu in WEBSITE > MENU, preferrably as a submenu under the FORMS menu.