Which likelihood function is used in linear regression?

Linear regression is about how $x$ influences/varies with $y$, the outcome or response variable. The model equation is $$ Y_i =beta_0 + beta_1 x_{i1} + dotsm + beta_p x_{ip}+epsilon_i $$ say, and how $x$ is distributed doesn't by itself give information about the $beta$'s. That's why your second form of likelihood is irrelevant, so is not used.

See Definition and delimitation of regression model for the meaning of *regression model**, also What are the differences between stochastic v.s. fixed regressors in linear regression model? and especially my answer here: What is the difference between conditioning on regressors vs. treating them as fixed?

As you say, in this case the choice doesn't matter; the maximization over $w$ is the same. But you touch on a good point.

Typically for regression problems we prefer to write the version with the conditioning on $x$ because we think of regression as modeling the conditional distribution $p(y | x)$ while ignoring the details of what $p(x)$ may look like. Succinctly, we only want to model how $y$ responds to $x$, and we don't care about how $x$ itself.

To see why, consider the model $Y = f(X) + varepsilon$, where $varepsilon sim mathcal{N}(0, sigma^2)$, we want the estimate function $hat{f}$ we produce to be as close to $f$ as possible (on average, and measured by e.g. least squares).

So suppose we have known data $(x_1, f(x_1)), ..., (x_N, f(x_N))$. Ultimately we just want to pull $hat{f}(x_1), ..., hat{f}(x_N)$ close to $f(x_1), ..., f(x_N)$. It doesn't matter the distribution $p(x)$ of how the inputs are scattered, as long as $hat{f}$ is close to $f$ on those values ($f$ being how $y$ "responds" to $x$).

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Here's how the math works out. From an MLE standpoint the goal is to get our likelihoods $P(x,y|w)$ as large as possible, but as you say, we are not modeling $P(x)$ through $w$, so $P(x)$ factors out and doesn't matter. From a fitting standpoint, if we want to minimize expected prediction error over $f$, $$min_f text{EPE}(f) = min_f mathbb{E}(Y - f(X))^2$$ omitting some computation we obtain the minimizing $f$ to be $$f(x) = mathbb{E}(Y | X = x)$$ so for least squares loss, the best possible $f$ depends only on the conditional distribution $p(y | x)$, and estimating it should not require any additional information.

However, in classification problems (logistic regression, linear discriminant analysis, Naive Bayes) there is a difference between a "generative" and a "discriminative" model; generative models do not condition on $x$ and discriminative models do. For instance, in linear discriminant analysis we do model $P(x)$ through $w$---we also do not use least squares loss in those.

That's a good question since the difference is a bit subtle - hopefully this helps.

The difference between in calculating $p(y|x,w)$ and $p(y,x|w)$ lies in whether you model X as being fixed or random. In the first, you're looking for the joint probability of X and Y conditioned on w, whereas on the 2nd one, you want to determine the probability of Y conditioned on X and w (usually denoted as $beta$ in statistical settings).

Usually, in simple linear regression,

$$Y = beta_0 + beta_1 X + epsilon$$

you model X as being fixed and independent from $epsilon$ which follows a $sim N(0,sigma^2)$ distribution. That is, Y is modeled as a linear function of some fixed input (X) with some random noise ($epsilon$). Therefore, it makes sense to model it as $p(y|X,w)$ in this setting as X is fixed or modeled as constant, which is usually denoted explicitly as $X=x$.

Mathematically, when X is fixed, p(X) =1 as it is constant. Therefore

$$p(y,X|w) = p(y|X,w)p(X) implies p(y,X|w)= p(y|X,w)$$

If X is not constant and given, then p(X) is no longer 1, and the above doesn't hold, so it all comes down to whether you model X as being random or fixed.