I have a brand new bar magnet rated as 1.23 Tesla by the manufacturer. However when I measured it with a meter, it gave a reading of 0.61 Tesl. Why?

I wonder how you measured it.

And at what distance is B supposed to be 1.23? 1m? 10 light-years?

Let's say the Br of a magnet is 1.23 Tesla. If I put my probe touching the magnet, it reads 0.6 Tesla. is there something wrong? Does it mean Br of 1.23 Tesla is equivalent to 0.6 with a probe?

The 1.23 would be the field between two such bar magnets, which equals the B at the center of one bar magnet.

B at the end of a bar magnet is one half the B at the center, so you measure one half of 1.23.

Does it mean that Br of a magnet is actually the theoritical B in the centre of a magnet? The B on the surface of a magnet is about half of the Br, for all configurations of a magnet? Where can I find the proof?

For a bar magnet, the proof is very simple. It also works for a solenoid.

Place two bar magnets touching end to end (N facing S). The B at their interface comes equally from each magnet, and so is twice what you would have if one magnet is pulled away.

Maybe I am using the Hall probe to measure magnetic force wrongly. Should I put the tip or the long side of the probe on the magnet surface?