> who is stationed permanently onsite at the client
Not really necessary. The distinction between FDEs and product engineers is that the FDE work is focused on a single deployment, whereas a product engineer works on a single product and is once-removed from actual clients.
Where the FDE spends time and what products he works on depends on the client requirements.
A business cultivation engineer is basically a sales engineer responsible for interfacing with and being a technical resource for the client's technical buying team (e.g. IT department).