Short Answer
Overall, if you want help with a practical need, you should provide details of what you're trying to accomplish and why you want to accomplish it. If you're simply interested in the theory, then the answer at the level of detail you've provided is "it depends."
Long Answer
The details matter. For instance:
- Windows doesn't like being installed on logical partitions, so converting a Windows installation from a primary partition to a logical partition will result in it becoming unbootable. (I gather that there are ways to work around this limitation, but AFAIK they all involve a smaller primary partition or more complex convolutions. These might be awkward to undo.)
- Converting a partition from primary to logical form, or vice-versa, can be done in a trivial way in some cases; but in other cases it will require resizing partitions, moving partitions, or converting other partitions, too. My FixParts tool can do the conversion, but not any prerequisite moving or resizing; and it will not guarantee a bootable OS afterwards. Some other Windows tools can do similar conversions, but I don't know the details of how much of the task they can handle.
- There are likely to be OS-specific quirks. Linux tends not to care much about primary vs. logical partition status. My memory of more obscure OSes (FreeBSD, OS/2, BeOS, etc.) is fading as I've not used them in such a long time, but I recall that DOS is limited like Windows.
- Depending on the OS and boot loader, you might need to re-install the boot loader after one or both of the conversions.
- A low-level full-disk backup is one trivially simple way to ensure you can restore to your starting point; however, such a backup is likely to consume a lot of space on the backup medium.
In the BIOS world, "boot information" is stored in all sorts of locations, such as:
- The Master Boot Record (MBR; the first sector of the disk)
- Sectors immediately following the MBR
- The Partition Boot Record (PBR; the first sector of an OS's boot partition)
- The BIOS Boot Partition (on GPT disks only, so irrelevant for your question, but I'm noting it for completeness)
- Files in various partitions.
BIOS-mode booting is increasingly irrelevant, since most computers sold since late 2011 use EFIs, not BIOSes, and EFI boots in an entirely different way. Your question notes primary and logical partitions, though, which implies the computer is likely to be booting in BIOS mode. If it's booting in EFI mode, though, it would likely use GPT, which doesn't have the primary/extended/logical partition distinction. Under GPT, boot loaders are stored as files on the EFI System Partition (ESP), rather than scattered about in the MBR, PBR, etc. If the computer is booting in EFI mode from an MBR disk, it probably doesn't care much about primary vs. logical partition status, although if you convert the ESP from primary to logical or vice-versa, that might confuse the EFI enough that the system wouldn't boot. This is something I've never tried -- and even if I had, the results might vary from one EFI to another, so I'd be reluctant to generalize about it.
Some boot loaders rely on partitioning details like a partition's number, primary/logical status, and the presence of a "boot flag" (aka "active flag") on a partition. Details vary from one OS and boot loader to another. The point of this paragraph is that backing up "boot information" depends on knowing the details of the OSes involved, the partition layout, perhaps what boot loader(s) are in use, etc.