How to make my old hard drive non-bootable?

Your Disk 0 partition 1 (Storage E:) is still has the Active flag enabled.. This is the cause of the problem. You have to remove the active flag from that partition as follows:

1. Open up cmd as administrator.
2. Type diskpart and hit enter
3. Type list disk and enter
4. Type select disk 0 and enter
5. Type list partition and enter
6. Type select partition 1 and enter
7. Type inactive and enter
8. Type exit

This will alter the hard drive to remove the "bootable" flag and empty the mbr. I cannot guarantee success but your bios should skip the hard drive if it doesn't find any boot flag on this hard-drive and no mbr. If used wrongly or under special circumstances this might kill only parts of your mbr or other data, like partition scheme, too, so make a backup before.

1. Boot up a linux. (sorry, dear windows user, search the Web for removing boot flag and dd, dd basically writes zeros to the first 446 bytes which are AFAIK reserved for the Mbr.)
2. Get the /dev/sdX number (probably lsblk or such, I will use /dev/sdh for now)
3. Run fdisk /dev/sdh where sdh is is obviously your right disk.
4. p your partition scheme, look for the partition number (first column, sdXY, ie. sdh6) and if the boot flag (second column) is checked with a *
5. a and then the partition number. NOTE: fdisk doesn't want you to enter sdh6 or even /dev/sdh6, it just wants 6 in this case.
6. p and check the results.
7. w to write changes and exit.
8. Additionally you should empty the mbr

Command to delete mbr only

The following command will erase mbr, but not your partitions:

dd if=/dev/zero of=/dev/sdc bs=446 count=1

Source: http://www.cyberciti.biz/faq/linux-clearing-out-master-boot-record-dd-command/ (not tested, but should work)

If that doesn't work and the old hard drive is still getting booted, you might try the hack to remove the /boot directory (or empty the partition, if so) on this hard drive. That is the place where the pc looks at boot time and where grub lies. Unfortunately you will loose your kernel (/boot/vmlinuz* mostly) and so but if you really mean to never boot up that device, you can try it. (if you only remove Mbr and bootable flag (or just grub), the boot repair tool can help you. If you delete your kernel and such, you have to reinstall Linux.

You should be able to enter the BIOS at startup, by pressing one of the F-keys, often F11. Some BIOS systems tell you at boot time, which F-key it is. Find the BOOT section of the menu, and there you can set your SSD as the first option.

You may want to set Disk 0 as the second option, if you want to be able to enter the boot menu of your BIOS, and boot manually into your Linux system. I advise you to keep Disk 0 bootable, otherwise you won't be able to boot to your Linux OS.

The new boot order should stay the same at next startup, and only change if you enter the BIOS again and make new changes.