Let $\mathbb T$ be the set of all complex numbers of modulus 1 (i.e the unit circle on the complex plane). For any element $\varphi$ of the complex Hilbert space $H$, the set $\mathbb T\varphi$ is called a ray (see here).
What is the intuition behind this name? Why a ray?
The source of this is that there are two equivalent philosophies to handle the state vector in quantum mechanics.
One philosophy is to not require the vector to be normalized. In that case, multiplying a state vector with an arbitrary non-zero complex number gives another vector representing the same state. The set $\{c\psi\mid c\in\mathbb C^\times\}$ is called a ray; the intuition is that it is similar to a ray in real vector spaces, defined as $\{\lambda v\mid \lambda\in\mathbb R_{>0}\}$.
However in that philosophy, you always have to include explicit normalization factors when calculating probabilities, therefore also another philosophy exists where the state vector is required to be normalized. Now normalization only fixes the absolute value of $c$ in the ray description above, not the phase factor $\mathrm e^{\mathrm i\varphi}$. Therefore in this philosophy, the set of state vectors that describe the same state is the set you described; it is the intersection of a ray with the unit sphere.
However in my many years of doing quantum mechanics I've never seen that set to be called a ray. Rather in that form, one usually speaks of the free choice of phase.