The field for procurement (make to order / make to stock) only tells you if you should procure for a specific sale order or in general because you need the product on stock. The other field for procurement (Buy / Manufacture) tells you HOW the product will be procured (how you will "get" it). So, if that is set to Buy then Odoo will create a purchase order as a reaction. If it is set to Produce then Odoo will create a manufacturing order according to the underlying Bill of Material. By the way, I did not see the field to choose either in you screenshot. Only the routes field which is not something that I have used before.

This is the on-line version of Odoo, I guess it is not quite ready for production? It is hard to understand how the procurement works

I cannot imagine the online version not being production ready in that sense, but it might be. Maybe this piece of documentation will make it more clear on how to use/configure Odoo: https://doc.openerp.com/book/5/5_14_Stock/5_14_Stock_inv/#automatic-procurement

The documentation is not applicable to the online version anymore, the flow changed a bit, at least there is no supply method and the "Routes" option is new. And Fabien's answer didn't work well, with 'Routes' set to MTO, even though there is a reordering rule, PO won't be generated. However, if 'Routes' is set to 'Buy', then PO will be generated. Moreover, looks like it is necessary to manually run the scheduler to trigger the creation of PO, I have installed mrp_jit, not sure what else I should make it automatic.