As Prahar mentioned, operators that transform as a highest weight of the "X algebra" are called "X primary". So in non-supersymmetric 2d CFTs, where you have the global algebra ($SL(2, \mathbb{C})$) and the local algebra (Virasoro), there are two types of primary operators. Quasiprimary is a synonym for global or $SL(2, \mathbb{C})$ primary. If someone just says primary, this is a synonym for local or Virasoro primary. In supersymmetric 3d CFTs, for example, you also have two types. Conformal primaries generate a representation of the 3d conformal algebra $SO(4, 1)$ but there are also superconformal primaries for representations of the full superalgebra $OSp(\mathcal{N} | 4)$. Supersymmetric 2d CFTs would then lead to four natural types of multiplets. The largest are super-Virasoro, the smallest are regular-global and in between you have regular-Virasoro and super-global.
Another point is that the highest weight definition implies the (scalar) transformation law (1). A global primary, if it's a Lorentz scalar, has
\begin{align}
[D, \phi(0)] = \Delta \phi(0), \quad [M_{\mu\nu}, \phi(0)] = 0, \quad [K_\mu, \phi(0)] = 0
\end{align}
as the "spin part" of its infinitesimal conformal transformation. Since $P_\mu$ generates translations, you can use this and the conformal algebra to solve for the "orbital part" as
\begin{align}
& [P_\mu, \phi(x)] = \partial_\mu \phi(x), \quad [M_{\mu\nu}, \phi(x)] = x_{[\nu}\partial_{\mu]} \phi(x), \quad [D, \phi(x)] = (x\cdot\partial + \Delta)\phi(x) \\
& [K_\mu, \phi(x)] = (2x_\mu x\cdot\partial - x^2\partial_\mu + 2\Delta x_\mu)\phi(x).
\end{align}
These infinitesimal transformations can in turn be exponentiated into the finite one you wrote above. As you point out, there would have to be more terms if you wanted (1) to work for the conformal transformations in 2d which are not global. The simplest demonstration of this is the transformation law for the stress tensor.
\begin{equation}
T^\prime(z^\prime) = \left ( \frac{\partial z^\prime}{\partial z} \right )^{-2} \left [ T(z) - \frac{c}{12} \{ z^\prime, z \} \right ]
\end{equation}
For holomorphic transformations not in $SL(2, \mathbb{C})$, the anomalous (Schwarzian derivative) term is non-zero.
So to answer your questions,
If $x^\prime(x)$ is a global conformal transformation, then (1) holds for all quasiprimaries, of which (Virasoro) primaries are a special case. If $d = 2$ and $x^\prime(x)$ is one of the other conformal transformations, it does not hold.
Virasoro primaries do not exist in $d > 2$. But $[L_1, \phi] = [\bar{L}_1, \phi] = 0$ becomes $[K_1, \phi] = [K_2, \phi] = 0$ after taking linear combinations. So global primaries have the $[K_\mu, \phi] = 0$ definition both in $d = 2$ and $d > 2$.
The biggest time saver in these theories is that states can be grouped into multiplets related by symmetry. So the spectrum of your theory is like a tree where the trunk is a primary and the branches are its descendants. Among the branches, you can also find infinitely many "subtrees" which are spawned from a quasiprimary.
