My numerical analysis book uses the same definition for significant digits as stated in this post Why does relative error give number of correct digits?
My question is how can I tell when the number of significant digits that $x$ and $y$ agree on is exactly $t$ or $t+1$? I can't think of a case where they agree on $t+1$ significant digits
Let's suppose $\beta=10$ so we are working with decimal numbers.
and as an example suppose $y=1.0002\, x$ for $x>0$.
Then $\frac{|x-y|}{|x|}=0.0002$ and $10^{-4} < 0.0002 \le 10^{-3}$.
If $x=8$ then $y=8.0016$ which agrees with $x$ to $3$ significant digits.
If $x=2$ then $y=2.0004$ which agrees with $x$ to $4$ significant digits.
Whether it is $t$ or $t+1$ significant digits depends on the circumstances of the case, but if the first significant digit of $x$ and the first significant digit of $\frac{|x-y|}{|x|}$ both being small makes $t+1$ more likely