The fields transform under finite conformal transformations as$^1$
$$
\Phi^a(x') \mapsto {\Phi^a}'(x) = \Omega(x')^\Delta\,D(R(x'))^{\phantom{b}a}_b \,\Phi^b(x')\,. \tag{1}\label{main}
$$
as given in equation $(55)$ of $[1]$. Let's break it down:
- $\Delta$ is the conformal dimension of $\Phi$.
- $\Omega$ is the conformal factor of the transformation.
- $D$ is the spin representation of $\Phi$.
- $R$ is the rotation Jacobian of the transformation
So let's compute these things. The spin and the conformal dimension $\Delta$ are given. The first thing we have to look at is the Jacobian.
$$
\frac{\partial x^{\prime \mu}}{\partial x^\nu} = \Omega(x') R^\mu_{\phantom{\mu}\nu}(x')\,.
$$
This implicitly defines both $\Omega$ and $R$ and it is not ambiguous because we require $R \in \mathrm{SO}(d)$, namely
$$
R^{\mu}_{\phantom{\mu}\nu} \,\eta^{\nu\rho}\,R^{\lambda}_{\phantom{\lambda}\rho} \,\eta_{\lambda\kappa}= R^{\mu}_{\phantom{\mu}\kappa}\,.
$$
You can immediately see that for the Poincaré subgroup of the conformal group $\Omega(x')= 1$, whereas for dilatations $\Omega(x') = \lambda$ and for special conformal transformations
$$
\Omega(x') = \frac{1}{1+2(b\cdot x') + b^2 {x'}^2}\,.\tag{2}\label{omega}
$$
This can be proven with a bit of algebra. Using your definition of SCT
$$
{x'}^\mu = \frac{x^\mu - b^\mu x^2}{1+-2(b\cdot x) + b^2 {x}^2}\,,
$$
one can check
$$
\frac{\partial x^{\prime \mu}}{\partial x^\rho} \eta_{\mu\nu} \frac{\partial x^{\prime \mu}}{\partial x^\lambda} = \frac{\eta_{\rho\lambda}}{(1-2(b\cdot x) + b^2 x^2)^2}\,.
$$
That means that the Jacobian is an orthogonal matrix up to a factor, which is the square root of whatever multiplies $\eta_{\rho\lambda}$. Then we have to re-express that as a function of $x'$. After some algebra again one finds that it suffices to change the sign to the term linear in $b$.
Finally, how does one compute $R$? Well, it's just the Jacobian divided by $\Omega$. In the case of special conformal transformations one has (there might be mistakes, redo it for safety)
$$
R^{\mu}_{\phantom{\mu}\nu} = \delta^\mu_\nu + \frac{2 b_\nu x^\mu - 2 b^\mu (b_\nu x^2+ x_\nu - 2 (b\cdot x) x_\nu) -2b^2 x^\mu x_\nu }{1-2b\cdot x +b^2 x^2}\,,
$$
which, as before, needs to be expressed in terms of $x'$.
If you are interested in $\Phi$ scalar then $D(R) = 1$ and you can just plug \eqref{omega} into \eqref{main} to obtain the transformation. If you want to consider also spinning $\Phi$ then it's not much harder.
For spin $\ell=1$ the $D$ is just the identity, namely
$$
D(R)^{\phantom{\nu}\mu}_\nu = R^{\phantom{\nu}\mu}_\nu\,.
$$
For higher spins one just has to take the product
$$
D(R)^{\phantom{\nu_1\cdots \nu_\ell}\mu_1\cdots \mu_\ell}_{\nu_1\cdots \nu_\ell} = R^{\phantom{\nu_1}\mu_1}_{\nu_1}\cdots R^{\phantom{\nu_\ell}\mu_\ell}_{\nu_\ell}\,.
$$
Again, by plugging these definitions in \eqref{main} you obtain the desired result.
$\;[1]\;\;$TASI Lectures on the Conformal Bootstrap,
David Simmons-Duffin, 1602.07982
$\;{}^1\;\;$The way the transformations are written in the lecture notes linked above differs a bit from Di Francesco Mathieu Sénéchal. The difference is that Di Francesco et al. make an active transformation $x \to x'$ with
$$
\Phi(x) \mapsto \Phi'(x') = \mathcal{F}(\Phi(x))\,,
$$
while David Simmons Duffin makes essentially the inverse transformation $x' \to x$
$$
\Phi(x') \mapsto \Phi'(x) = \mathcal{F}^{-1}(\Phi(x'))\,.
$$
That is why in the above discussion the indices of $R^\mu_{\phantom{\mu}\nu}$ get swapped when passed inside $D$ as $D(R) = R^{\phantom{\nu}\mu}_{\nu}$. And that's also why we get a factor $\lambda^\Delta$ rather than $\lambda^{-\Delta}$ as Di Francesco et al. This is all consistent as long as it is clear what one is doing.
Best Answer
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.