Windows – Multiple recovery partitions in Windows 10

First, ignore Windows' identification of your logical partitions as "primary;" that's a long-standing Windows bug.

Second, it's conceivable that the NTFS partition you created in Linux has the wrong partition type code. You can view the type codes using the Linux fdisk command, as in fdisk -l /dev/sda (typed as root or preceded by sudo). An NTFS partition should have a type code of 07 under the Id column in fdisk's output. If it doesn't have that code, you can change it with fdisk; type fdisk /dev/sda, then use the t option to change the type code and w to save your changes.

Another possibility that occurs to me is that the NTFS structures may be invalid. You could try converting the partition to FAT and see if it's visible then; and if it is, convert it back to NTFS in Windows. That might produce an NTFS that Windows would like.

The only other possibility that springs to mind is that this is a Windows bug -- namely, that it can't read a primary partition that comes after an extended partition. If so, the obvious (but awkward) solution is to delete the NTFS partition, use a Linux emergency disc and GParted to move your Linux partitions to the end of the disk, and to create a fresh primary NTFS partition that comes before the extended partition. Before you try this, though, I recommend you wait and investigate it some more; somebody else may come up with another (safer and easier) thing to try.

Whatever you do, though, do not try to create a new partition in Windows; the Windows partitioner has the annoying habit of converting disks to use LDM (aka "dynamic disks") whenever you want more than four partitions. Linux can't boot from such setups (or if it can, doing so is a very poorly documented), so if Windows were to do such a conversion you'd be in even deeper trouble. I don't know offhand if Windows would do this when the disk contains non-Windows partitions, and I don't think trying it on your disk is a good way to learn about this!

See Lenovo's article titled Understanding hard drive partitions on Lenovo systems with Microsoft Windows 7 and Windows 8 - ThinkCentre, ThinkStation, ThinkPad:

For Microsoft Windows 8, which requires three partitions, Lenovo created the Recovery Partition to allow the recovery of the initial operating system and software on the computer to the same state that they were in before the computer was activated. ... The far left small Recovery Partition, EFI System Partition and Windows 8 operating system partition, is required by Windows 8. The larger Recovery Partition is the Lenovo created partition.

There's also an explanation of Lenovo's partitioning scheme in a response to New G580 has seven partitions, Help identify? on the Lenovo support forum:

1. WINRE_DRV 1000MB OEM - this is the Windows Recovery Environment - this has it's own interface / it doesn't need the 7th partition to operate
2. SYSTEM_DRV 260MB System - check this thread
3. LRS_ESP 1000MB OEM- The third partition is the EFI System partition which is used for data storage for the Extensible Firmware Interface (EFI) interface
4. (blank) 128MB MSR - The fourth partition is the Microsoft Reserved Partition which is also used for data storage for the Globally Unique Identifier (GUID).
5. Windows8_OS 884.2GB Primary - Windows 8 OS
6. LENOVO 25GB Primary - this contains the Application and Drivers that are preloaded on the OS (the drivers listed on this folder are also available on the lenovo website)
7. PBR_DRV 20GB OEM - this is your recovery partition - you will need to press the One Key Recovery button to activate this partition.