I think you misinterpreted the installer's prompt; I don't think it was telling you to create a separate
/boot partition (which uses a Linux filesystem and holds the Linux kernel and related files), but to create a BIOS Boot Partition, which holds part of the BIOS-mode version of GRUB without using a filesystem at all. The two are totally different things. Some installations require one or the other of these partitions, and some require both. Your computer has both of them, but in fact, neither is actually necessary.
Also, some of your partition sizes are way out of line:
/dev/sda6 is your
/boot partition, and is 488 MiB, which is a decent size. (~500 MiB is common these days.)
/dev/sda7 is your swap partition, and is 15 GiB, which is probably not too bad. In the past, swap was routinely 1-2x the amount of installed RAM. Today that's normally overkill, unless you want to suspend to disk, in which case it should be at least as big as your RAM. If you don't want to suspend to disk, half your RAM size is probably adequate. Some people favor using no swap space at all, but it does have its advantages even if you don't use all your RAM for programs, since the kernel can swap out less-used temporary data in favor of caches that can speed up disk access.
/dev/sda8 is your root (
/) partition, and is 171 GiB, which is much too big if you've got a separate
/home, as you do. In most cases, ~20 GiB is enough for an Ubuntu root (
/) partition. For a big installation with lots of programs, 50 GiB should be more than adequate. (OTOH, if you don't use a separate
/ should be the bulk of your Ubuntu-related storage -- the size of both
/home if they were separated.)
/dev/sda9 is your BIOS Boot Partition, and is 122 MiB, which is way too big for this partition. 1 MiB is normally sufficient -- but if you install in EFI mode, this partition is completely unnecessary.
/dev/sda10 is your
/home partition, and is 855 MiB, which is much too small. Your user files reside in
/home, so it should be big enough to hold whatever you plan to store there -- word processing document, Web browser caches, MP3 files, videos, etc. Some of these can be huge. Thus, on a desktop or laptop with a separate
/home, this is normally the largest partition.
The best way for you to proceed is to re-install Ubuntu as follows:
- In your firmware setup utility, disable the Compatibility Support Module (CSM), aka legacy support or BIOS-mode boot support.
- You may need to re-create your Ubuntu installation medium (or maybe not). The key is to ensure that it includes an EFI-mode boot loader. See this page of mine for more on the CSM and its problems and how to create a suitable boot medium (the last is near the bottom, in "Creating EFI-Bootable USB Drives.") It's unclear to me if your current medium has the EFI-mode boot support.
- Boot the Ubuntu installer, but select the "try before installing" mode.
- Open a Terminal window and verify that you've booted in EFI mode by looking for a directory called
/sys/firmware/efi. If it's present and contains various files and subdirectories, you've booted in EFI mode; otherwise, you've booted in BIOS/CSM/legacy mode, and you should review the preceding steps and fiddle with your settings or USB drive until you can boot in EFI mode.
- Delete the partitions corresponding to your current installation --
- Re-run the Ubuntu installer. You can either create partitions manually, but along more sensible lines; or you can tell the installer to install in the disk's (now) unpartitioned space. I'd recommend a 20-to-50 GiB root (
/), a swap partition of roughly its current size (adjust up or down as you see fit; see my earlier comments), and the rest as
/home. There's no need for a separate
/boot partition or a BIOS Boot Partition. Be careful with partitioning, though; people have accidentally wiped out their Windows installations by selecting the wrong option!
When the installer finishes installing, the computer should reboot into GRUB, which should give you the option of booting either Ubuntu or Windows.