I just want to use the Recovery HD to repair the disk ... I do not want to recover the whole drive from internet recovery or reinstall as new.
It's not giving you that choice, something went wrong with RecoveryHD (or it doesn't exist, like if you used Superduper to reverse clone with) so it's booting into Internet Recovery which is almost the same thing.
Both do the same thing, but IR installs the OS X verison that came with the machine 10.7 and up, and RecoveryHD installs the current upgraded OS X verison.
If you tried booting into RecoveryHD and it instead went to Internet Recovery and if diskutil list only shows a EFI partition, then something seriously BAD happened to your boot drive.
It should look almost like this, except last partition #3 will be labeled "OS X base system" or something like that.
(I'm running Snow Leopard which doesn't need RecoveryHD and I have another OS X boot partition, cloned for backup/restore)
I see that I have an EFI boot partition available.
You can't boot from that, it's a software based firmware that loads before any operating system does so it enables your at boot key commands and other things regardless of what operating system loads.
In fact if you reboot the machine holding Command V down, you can watch EFI load first.
How do I get it to boot into the local Recovery HD and NOT internet recovery?
Use the Internet Recovery to Reapir Disk on your entire disk0 also Repair Permissions (repeat 2x) and reboot, hopefully that will fix it or your doing the next step below.
So the problem is you likely need to grab files off the drive first, because your going to have to set it back up by reformatting it, or replace the drive if it's defective.
Create a data recovery/undelete external boot drive
Can you provide more detail what diskutil list actually shows?
Does Disk Utiltiy give any indication of the SMART status?
What were you doing that prompted this to occur? Any clues you can share?