Apple - Unable to install OS X, cmd+R opening internet recovery instead of usual recovery

Your model Mac was shipped with Yosemite (OS X 10.10) installed. The tags you placed on the question suggest you have upgraded to El Capitan (OS X 10.11). There is the possibility you could have erased the Recovery partition by using the Disk Utility if you had Yosemite installed. Evidently, this would be much harder (or perhaps impossible) to do if El Capitan was installed. Also, I do not know if you were using Core Storage and if so, how this would effect the ability to erase the Recovery partition.

Evidentially, you can now only boot using OS X Internet Recovery. You can use Internet Recovery to install the version of OS X that came with your machine. From this, you can go to the App Store and download and install newer versions of OS X.

An alternative would be to use another Mac to download OS X. You can then transfer the download to a USB drive. You then use the USB drive to install OS X on your Mac. The instructions are given the Apple web site: Create a bootable installer for OS X.

Disconnect from the network if you care and then hold the option key when you reboot.

If you have a viable standard recovery HD it will be shown. From there you can attempt to repartition or download an installer.

• Check your time, if it’s off more than three minutes, TLS and SSL will sporadically fail.
• Then check DNS, if your Mac can’t find Apple’s servers, this won’t work.

Another option is to use another bootable OS to run the repair / install. Holding option will let you select those OS once they are connected.

Once you're booted - the steps are the same as recovering from Internet Recovery: Erased Macintosh HD and installing OS X Lion again

I have a workaround for this problem. It seems somehow my system hates my Wi-Fi connection (may be due to some encryptions related to Wi-Fi). When I went to my friend's place and connected to Wi-Fi, I was able to proceed with Internet Recovery. After waiting 5 minutes, I landed on a page similar to normal recovery mode and then I was able to install OS X normally (Yosemite was installed in this case).

If this doesn't work, I would have create a bootable installer for OS X as suggested by David's and bmike's answer