The thief

is right, basically because

the endpoints are not included.

The problem with one of the two suggested approaches is that

the cop predicts a definite point of convergence, since the thief has a finite amount of space in which to manoeuvre, but what if the predicted convergence is at the point $t=0$ or beyond? That means it will actually never be reached. The cop cannot continue moving at speed one for time $2/3$ or longer.

The space in which they're moving is sort of like a 1-dimensional hyperbolic space: they can get infinitesimally close to the boundary but they can never actually reach it.

TL;DR: it seems that, on an open line segment, Zeno's paradox is actually true.

The thief

is right, basically because

the endpoints are not included.

The problem with one of the two suggested approaches is that

the cop predicts a definite point of convergence, since the thief has a finite amount of space in which to manoeuvre, but what if the predicted convergence is at the endpoint $0$ or beyond? That means it will actually never be reached. The cop cannot continue moving at speed one for time $2/3$ or longer.

The space in which they're moving is sort of like a 1-dimensional hyperbolic space: they can get infinitesimally close to the boundary but they can never actually reach it.

TL;DR: it seems that, on an open line segment, Zeno's paradox is actually true.

The thief

is right, basically because

the endpoints are not included,

although I agree with Ross Millikan that the problem isn't fully defined.

To make this ill-definedness clear, what happens if the cop and the thief both move leftwards at speed 1? At time $t=1/3$, the cop is at the thief's starting position, and the thief is where? We know where the thief is at any time $t=\frac{1}{3}-\varepsilon$, but not at the time $t=1/3$ itself.

The problem with the cop's approach is that

he predicts a definite point of convergence, since the thief has a finite amount of space in which to manoeuvre, but what if the predicted convergence is at the point $t=0$ or beyond? That means it will actually never be reached.

TL;DR: on an open line segment, Zeno's paradox is actually true.

The thief

is right, basically because

the endpoints are not included.

The problem with one of the two suggested approaches is that

the cop predicts a definite point of convergence, since the thief has a finite amount of space in which to manoeuvre, but what if the predicted convergence is at the point $t=0$ or beyond? That means it will actually never be reached. The cop cannot continue moving at speed one for time $2/3$ or longer.

The space in which they're moving is sort of like a 1-dimensional hyperbolic space: they can get infinitesimally close to the boundary but they can never actually reach it.

TL;DR: it seems that, on an open line segment, Zeno's paradox is actually true.