Recover Grub dual boot UEFI Windows 8 + CSM installed Lubuntu 20.04

Question

I have an old Toshiba Satellite C855D-S5105. It has the original Windows 8. I have just installed dual booting Lubuntu on it, selecting "Something Else" during installation. To be able to boot the installation disk, I had to change boot from UEFI to CSM, as per this, which is what I am currently using.

After installation, there was no grub. I recovered it editing /etc/default/grub. But I don't have the Windows entry there.

Now I see here I might have done wrong

Case when Ubuntu must be installed in UEFI mode

Having a PC with UEFI firmware does not mean that you need to install Ubuntu in UEFI mode. What is important is below:

if the other systems (Windows Vista/7/8, GNU/Linux...) of your computer are installed in UEFI mode, then you must install Ubuntu in UEFI mode too.

(which, as mentioned above, I did not do). This also suggests I did wrong.

To get the Windows entry in grub, I tried this, but it did not work. So I tried this other answer to the same question:

$ lsblk 
NAME    MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT
sda       8:0    0 298,1G  0 disk 
├─sda1    8:1    0   450M  0 part 
├─sda2    8:2    0   260M  0 part 
├─sda3    8:3    0   128M  0 part 
├─sda4    8:4    0    64G  0 part /media/user1/TI10657600C
├─sda5    8:5    0    91G  0 part /media/user1/Data
├─sda6    8:6    0  10,4G  0 part 
├─sda7    8:7    0   3,8G  0 part [SWAP]
├─sda8    8:8    0    11G  0 part /
├─sda9    8:9    0  68,4G  0 part /home
└─sda10   8:10   0  48,8G  0 part /media/user1/02B990C61E8CA20B
sr0      11:0    1  1024M  0 rom

Description: sda1-3 (ntfs, fat32, ntfs) already existed. sda4 is my Windows partition. sda5 is my Windows data partition. sda6 is an existing recovery partition (which is actually at the end of the drive). sda7-10 were added by me during Lubuntu installation (swap, root, home, and an extra ntfs partition for exchange, I might later extend sda9 with this).

I see there is no /boot/efi partition. So now I am stuck, and I do not want to take any further action to avoid messing with my system (if I did not already do that).

I mean to have dual boot, be it UEFI (I guess it is preferrable) or not. So my questions are:

Can I get UEFI dual boot? What steps should I follow?
If not, can I get legacy dual boot? What steps should I follow?

Note that I can boot the Windows partition, entering the BIOS and setting UEFI boot. This boots straight to Windows. Then I can switch the BIOS to CSM boot, and that goes straight to grub (where there is no Windows).

Related (but which I did not risk trying yet):

Recover Grub dual boot UEFI Windows 8 + CSM installed Lubuntu 20.04

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