ADE 4.5 does include seamless fulfillment, however, there is a caveat to this. It requires the fulfilling server running Adobe Content Server to be on version 6.5. If the provider that you're buying the book from isn't using this, then your devices won't see the book in the cloud. Then the eReader that you're using must also be using RMSDK 12.5. As far as I know, only ADE 4.5 is using RMSDK 12.5. So it's very likely that the eReader on your device isn't using that version yet (most 3rd party eReader providers are slow to update).
To recap: ADE and/or your 3rd party eReader app and ACS must be on specific software levels to have seamless fulfillment work. If this isn't the case then this won't work for you. You should be able to ask your eBook provider which version of ACS they are on. If it's not 6.5, then seamless fulfillment is not something that you'll be able to take advantage of.
Thank you for the information.
In my case, the e-book provider is Overdrive (library books). Does this mean I would have to request Overdrive to update their Adobe content server to version 6.5?
Also, is it possible for my current e-reader to be upgraded to version "RMSDK 12.5" through a software update?- Or is this version only possible with future e-reader models?
You could certainly ask Overdrive if they have plans to upgrade to 6.5. They may already have plans or they may have no intention of doing so. The capability for seamless fulfillment for your eReader would need to be pushed out in software update by the company that develops it. It's the same thing there as with Overdrive; they may or may not integrate it.