How persistence works with TC is probably different to many (if not most) other distributions. Instead of creating a "full" installation of the OS on the root disk (often referred to here as "scatter install), it uses a memory based approach were the initial (minimal) system is created (in memory) from the initrd file (i.e. 'tinycore.gz'). In addition to this the user may choose a set of extensions (which are SquashFS archives) that augment the initial system by being loop mounted and integrated into the running file system.
Persistence in the case of TC means to ensure that the selected extensions, as well as configuration files and other user data (saved into a archive file, i.e. 'mydata.tgz') are available again after a reboot. You really have to read the
concepts page carefully. It goes into more detail and shows some possible variations to the typical setup.
When performing a USB installation of TC a few things happen: A boot loader gets installed, plus the kernel (i.e. 'bzImage') and the initrd are stored on the USB drive. Further provisions are made for the storage of extensions (i.e. *.tcz files) and the backup archive.
The backup options you see as you exit from TC are referring to whether or how to perform the backup before shutdown. There is no relationship between these options and how extensions are handled at the next boot time.
You are right insofar as the green color in the mount tool is indication for a mounted device. Your assumption that '/dev/sdb1' is your USB drive sounds reasonable, but it's not a certainty. To find out more about a device here are a couple of commands (to be run within a terminal) that should help to shed some light into this matter:
- dmesg | grep sdb provides you with the relevant information from the boot process, e.g. the device recogition,
- fdisk -l /dev/sdb shows the available partitions, and
- blkid /dev/sdb1 shows some information about the individual partition, e.g. typically the UUID and / or label, plus the file system type
Having all those details should allow you to confirm whether '/dev/sdb1' is your USB drive or not.
EDIT: Hmmm, a bit of an overlap as I did not realize that gerald_clark had posted meanwhile, but I'll leave it as it is.