True or false? Does this mean that tables and chairs, rivers and rocks, by virtue of the fact that they exist, can be called 'beings'?
Traditionally in philosophy, anything that can be said to be is a being
No. Philosophers made the distinction between things that really exist by opposition to things which only seem to us to exist, though they also immediately disagreed as to what are mere appearances and what are the true Beings.
This is also the route that science is following. Scientists try to get at the fabric of reality, the true Being. Everything else could then be regarded as the appearance to us of this fabric.
It seems that it is a sort of cognitive bias whereby we humans cannot stop searching for the actual beings which have to exist beyond appearances.
This is the only way. Starting from a state of near complete ignorance, the only way for humans to survive in an unforgiving world is to have this instinctive reflex to search for the truth. This is what we all do one way or the other.
Being belong not to the thing, but to thyself. You can say that the table is existing, but it is existing inside thy being - inside thy mind.
But what are things? The tables and the chairs, but Mountains and Rivers are not things. So, they are beings too, beings of "spirits". Water is not the thing, Water is being, but the water in bottle is the thing and it only existing.
If thou speakest with River - it is normal, looks like belief in someone spirit, but if you speak with a chair - it is abnormal.
You can own a house, but it is not being, only existing, home is being. If you have a house, but it is not thy home, it is a thing only, thy home can be far from the place you are live now. Your body is yours while you are alive, but thy being is out of your life.
Your question amounts to just semantics. Classifying something as a being is an invention made by humans. There is no correct answer here since humans mean different things even with the same word in different contexts.
However, if I had to guess, no, most people would not classify a chair for example as a being since a being usually implies consciousness.