When we say the universe had a beginning, do we mean the entire universe or the observable universe? Or did both of them have a beginning?
Generally speaking, when people talk about the universe having a beginning they're talking about the solution to the Einstein equations called the FLRW metric. This describes a universe that is homogenous and isotropic, and it seems to be a pretty good approximation to our universe.
The FLRW metric tells us how the density of the universe changes with time. NB it does not give the size of the universe. A homogenous isotropic universe is infinite in size by definition, and it remains infinite in size all the way back to the Big Bang.
Anyhow, if you wind time back to the Big Bang you find the density of the universe becomes infinite and the distance between any two points in the universe falls to zero, even though the universe remains infinite in size. This nonsensical result is why you'll often hear it said that GR can't explain the Big Bang, and it's why we expect some theory of quantum gravity to take over and prevent the density becoming infinite.
Anyhow, the point of all this ranting is that by definition the FLRW universe is the same everywhere. So if you believe the FLRW metric is a good approximation to our universe (and it seems to be) then the entire universe behaved the same as the bit we can see. The entire universe, including our bit, had a beginning.
However, this is a strictly classical perspective and does not describe phenomena like inflation that originate from quantum mechanics. The original theory of inflation didn't change the overall behaviour much, it just introduced a period of exponential expansion shortly after the Big Bang. But there is a more recent theory called eternal inflation that drastically changes our view of the universe's beginning.
In eternal inflation the universe has been inflating for all time and will continue to inflate for all time, so the universe as a whole doesn't have a beginning. However areas within the universe stop inflating and turn into the sort of slowly expanding spacetime that we see around us. These areas then appear to have a Big Bang, but this is simply the point at which inflation stopped for that particular area. Other areas would have Big Bangs that happened at a different time, while the universe as a whole had no beginning.
So the answer to your question is yes and no depending on which theory you believe. Unfortunately at the moment there is no experimental evidence to prove which answer is correct.
You should never distinguish between the universe and the observable universe. This is a fallacy of assuming that the word "exist" means something more than "something we can measure and interact with". There is no accepted or concievable way to explore other universe, so one must be very careful when talking about their existence. It is usually just a figure of speech.
Physics operates based on a philosophy called positivism, which is operational--- in order to ask a question you should give a prescription for how it is to be answered experimentally. If you can't, it isn't part of physics, at least not yet. In this case, it doesn't seem likely that there is any operational definition to this question.
The model of the universe that one uses should be bounded by the cosmological horizon, and this horizon came from an inflating small-size deSitter horizon (this is the inflation theory). The start of inflation is shrouded in mystery for now, but any attempt to extend the concept of the universe outside the cosmological horizon is at best unobservable, and most likely incompatible with the quantum gravitational holographic principle, which asks that the spacetime have a description along the horizon boundary.
The Penrose-Hawking theorem, strictly applied, only indicates if we have a trapped null surface, which we get with an enclosing boundary larger than the Hubble radius, there has to be a singularity or closed timelike curve sometime in the past in some subregion. Strictly speaking, not even the entire observable universe need to begin from a singularity.