It looks like your "system" ruby doesn't have the Rails gem installed. You probably don't want to be using the system ruby anyway. How about you try this:
rvm use 1.9.3 # switch to your Ruby 1.9.3 that the other app used
gem list # make sure rails is listed
gem install rails # (only if rails was not listed)
rails new myapp
There is nothing weird about the "weird thing" you observed. When you cd into a directory, RVM looks at files like .rvmrc, .ruby-version, and .ruby-gemset in that directory and it automatically changes your environment to match. You can run "rvm info" to see what kind of environment you are currently in (I usually focus on the GEM_HOME and GEM_PATH variables). In this case, cd'ing into your Rails app directory caused RVM to switch you to Ruby 1.9.3, and that's the Ruby where you had the rails gem installed. Seems normal to me.