The disk appearing to be directly attached is a function of the enclosure's driver and possibly controller, assuming it is a USB enclosure. If it were an eSATA connected enclosure I would generally expect it to appear as a directly attached disk.
Unfortunately there isn't a reliable way to change disks to being recognized as a removable disk without modifying the driver and possibly the firmware. Here's a link to some basic information on doing that, but it's clearly for device driver developers.
This may be something you can simply workaround though as Windows believing the disk is not removable is largely a cosmetic problem unless you want to disconnect the disk while Windows is running and the filesystem is mounted. If that is the case then the problem is an inconvenience requiring additional steps. If you want to disconnect while Windows is running then take the disk offline first through diskmgmt.msc. To do so right-click the disk (area to the left of the volumes) and select "Offline." All SATA disks are hot swap so it is OK to disconnect them as long as there are no mounted filesystems. You can also safely disconnect the disk by shutting down Windows (doing so will unmount the filesystem cleanly and power off the disk).
If you are unhappy with the way you access the external drive from a user interface perspective then consider these options.
First you can change the drive letter assigned to the filesystem through diskmgmt.msc. You can mount the filesystem into an empty folder like you would in Linux. You can even give the filesystem multiple mount points into empty folders. To do so, right click the filesystem in diskmgmt.msc and select "Change Drive Letter and Paths..."
Second you can create Favorites in Windows Explorer to access the root of the drive or nearly any number of sub-folders. To do so simply navigate to the desired location right click Favorites and select "Add current location to Favorites."
You got the partition table and formatting right the first time. For disks smaller than 2 TB MBR partition table is usually fine. For disks larger than 2 TB you will need to use GPT. This is true for pretty much any block device in a Windows and Linux environment. Given that you want to use the disk in both Windows and Linux choosing NTFS for your filesystem was a good choice because both operating systems can mount the NTFS filesystem read-write.
I would not recommend overwriting the partition table using DD, frankly I'm surprised doing that didn't cause problems.