Is finding the efficient frontier a max or min problem?

Question

I'm trying to understand where the efficient frontier comes from.

What I understand about the efficient frontier

I understand the efficient frontier is essentially a subset of the boundary of the investment opportunity set of attainable portfolios. Intuitively, it represents, for each point along the $\sigma_{p}^2$ axis, the highest return possible at that point, which carves out the subset of points seen in this diagram below.

Figure 1

MPT Optimization as a minimization problem

As I understand it, Modern Portfolio Theory has the following set up. Assume a portfolio is created through some combination of $m$ assets. The portfolio has a corresponding weight vector $\mathbf{w}\in [0,1]^m$ and we have a return vector $\mathbf{r} \in \mathbb R^m$. We also define $\boldsymbol{\mu}=E[\mathbf{r}], \boldsymbol{\Sigma} = \text{Cov}(\mathbf{r})$. ​ Then we have $\mu_p = \mathbf{w}^T \boldsymbol{\mu}$ and $\sigma_{p}^2 = \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}$.

Then we find the optimal $\mathbf{w}^*$ via

Optimization Problem 1

\begin{align*} \mathbf{w}^*&=\underset{\mathbf{w}}{\text{argmin }} \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w} \\ &\text{s.t.}\\ \mathbf{w}^T \boldsymbol{\mu} &= \mu^*\\ \sum_{i=1}^m w_{i} & = 1\\ \end{align*}

Source: https://en.wikipedia.org/wiki/Modern_portfolio_theory

MPT optimization as a maximization problem

According to this question,

Efficient frontier doesn't look good

The efficient frontier stems from the above optimization problem. But I see a problem with that.

Consider optimization problem 1.

Figure 2

If I slide along the $\mu_p$ axis for different expected returns, I will be able to now find the minimum variance for each one. But this would trace out that entire boundary, not just the efficient frontier. Therefore, it would seem to me this optimization problem would be better.

Optimization Problem 2

\begin{align*} \mathbf{w}^*&=\underset{\mathbf{w}}{\text{argmax }} \mathbf{w}^T \boldsymbol{\mu} \\ &\text{s.t.}\\ \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w} &= \sigma^{2*}_p\\ \sum_{i=1}^m w_{i} & = 1\\ \end{align*}

I feel like, although they are similar problems, the implications are different. For Optimization Problem 2, I imagine sliding along the $\sigma_p^2$ axis. For each one, I pick the max $r_p$. Intuitively, this would trace out the curve seen in the efficient frontier diagram above. This would be in contrast to Optimization Problem 1 which traced out the entire boundary, not just the efficient frontier.

My question

Are both formulation of the optimization problem for finding $\mathbf{w}^*$ equivalent?

If so, can you explain what conditions are necessary for Optimization Problem 1 to trace out only the efficient frontier and not just the entire boundary of the investment opportunity set of attainable portfolios?

If not, can you explain why the prior stack exchange question says Optimization Problem 1 describes the efficient frontier?