Consider a compact interval $[a,b]$. If $[a,b]$ had global coordinates, then there would be an homeomorphism $f:[a,b]\to U$ where $U$ is an open subset of $\mathbb{R}$ or an open subset of $[0,\infty)$.
But $U$ cannot be open in $\mathbb{R}$ because $U$ is compact, so $U$ must be open in $[0,\infty)$. But also in this case I think U must be a closed interval, precisely $U=$[min$f$,max$f]$. But this set cannot be open in $[0,\infty)$. So i deduce that $[a,b]$ does not have global coordinates.
But then in John Lee's book Introduction to smooth manifolds i read a statement of the form "Let $t$ denote the stadard coordinate on $[a,b]$".
So how should I interpret the statement above?
When I refer to the "standard coordinate" on a subset of $\mathbb R$ such as $[a,b]$, I just mean the standard coordinate of $\mathbb R$ restricted to the interval. You're right that it's not a "coordinate chart" in the sense that the term is defined for manifolds with boundary. Perhaps it's infelicitous to use the same term with two different meanings, but it would be far from the only time this happens in differential geometry!