I'm learning about Ito's calculus and SDE. If the following is a generic SDE:
$$ dx = \mu(x)dt + \sigma(x)dB_t$$
Can I consider the $dx$ as a quantity which describes the changes in $x$ due to a deterministic function $\mu$ plus white noise with $\sigma$ as variance?
Moreover, if $x$ were a stochastic process $X_t$, the previous equation becomes: $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$ Is it correct?
The answer to your question depends on the type of model you are dealing with, the underlying distribution of Brownian motion, the way you have expressed it, and nature of data you are dealing with. For example the OU stochastic differential equation $dT=dS+\gamma (S-T)dt+ \sigma dW$ can be expressed as $$d(T-S)=\gamma (S-T)dt+ \gamma ^2 \sigma dW$$
If $T$ is temperature and $S$ is mean, this form means that if you remove the seasonal component, through calculation of $T-S$, the residual $R=T-S$ will follow a probability distribution with mean $\gamma S$ and variance $\gamma ^2 \sigma ^2$.
Read the works of Gymmerah et al 2021, Primak et al 2001, Dzupire et al, Leobacha and Ngare (2011) for more understanding.
The most intuitive way for me to understand these equations is to simulate paths: $$\Delta x_i\equiv x_{i+1}-x_i=\mu(x_i)\Delta t+\sigma(x_i,t_i)\xi_{t+1},\, \xi\sim\mathcal N(0,1)$$
In each simulation $j$ we create a path: $x^{[j]}=[x_0,x^{[j]}_1,x^{[j]}_2,\dots]\\ x^{[j]}_1=x_0+\mu(x_0)\Delta t+\sigma\xi^{[j]}_1\\ x^{[j]}_2=x^{[j]}_1+\mu(x^{[j]}_1)\Delta t+\sigma\xi^{[j]}_2\\ \dots$
Infinite set of paths $x^{[j]}$ comprise the solution of the equation with parameters $\mu,\sigma^2$.
This highlights the difference with deterministic equations without the stochastic terms such as $\sigma dB$.
In your stochastic equation, only the variance is deterministic in sense that it only depends on observed $x_i$ at time $t_i=t_0+i\Delta t$. So, each increment $\Delta x_i$ has a conditionally (on $x_i$) deterministic and stochastic parts.
In a fully deterministic differential equation you have only one path of evolution, e.g. $$\Delta x_i\equiv x_{i+1}-x_i=\mu(x_i)\Delta t\\ x_{i}=x_0+\sum_{k=1}^i\mu(x_i)\Delta t\\ x^{[]}=[x_0,x_1,x_2,\dots]\\ x_1=x_0+\mu(x_0)\Delta t\\ x_2=x_1+\mu(x_1)\Delta t\\ \dots$$
