How to reconcile these two versions of a "linear model"? Outline
I am taking a course in which the professor, unless I'm badly misunderstanding something, is discussing two varieties of linear models.
Version 1

The general linear model is $\boldsymbol Y = \boldsymbol X
 \boldsymbol \beta + \boldsymbol \epsilon$. Written out, it is
\begin{align*} \begin{pmatrix} Y_1\\ \vdots\\ Y_n \end{pmatrix} = \begin{pmatrix} h_1(X_1) & \dots & h_p(X_1)\\ \vdots &  & \vdots\\ h_1(X_n) & \dots & h_p(X_n) \end{pmatrix} \begin{pmatrix} \beta_1\\
 \vdots\\ \beta_p \end{pmatrix}
 + \begin{pmatrix} \epsilon_1\\ \vdots\\ \epsilon_n \end{pmatrix} \end{align*}
Written out, this takes the form
\begin{align*} Y_i = \beta_1 h_1(X_i) + \dots + \beta_p h_p(X_i) + \epsilon_i \end{align*}
for $i = 1, \dots, n$.

Version 2

\begin{align*} \begin{pmatrix} Y_1\\ \vdots\\ Y_n \end{pmatrix} = \begin{pmatrix} X_{11} & \dots & X_{1p}\\ \vdots &  & \vdots\\ X_{n1} & \dots & X_{np} \end{pmatrix} \begin{pmatrix} \beta_1\\ \vdots\\ \beta_p \end{pmatrix} + \begin{pmatrix} \epsilon_1\\ \vdots\\ \epsilon_n \end{pmatrix}. \end{align*}
Or written out,
\begin{align*} Y_i = X_{i1} + \dots + X_{ip} + \epsilon_i.
 \end{align*}
for $i = 1, \dots, n$.

It took me a while to realize that these two are quite different, because in the first one the rows of $\boldsymbol X$ are functions of the same explanatory variable $X_i$. I guess you could say that Version 1 only involves one explanatory variable. Version 2, on the other hand, looks more like a regular setup for multiple regression, with each $Y_i$ being written as a linear combination of the $i$th observations of all $p$ predictors.
How do I reconcile these? I guess you could combine them by making $\boldsymbol X$ $p^2$ columns wide and $\boldsymbol \beta$ $p^2$ rows long, by applying the functions $h_1, \dots, h_p$ to all $p$ explanatory variables in each row of $\boldsymbol X$? (I suppose there is no reason why the number of functions $h$ needs to be equal to the number of explanatory variables $X$.)
In other words, if I wanted to combine these two paradigms, would I be looking at something like

\begin{align*} \begin{pmatrix} Y_1\\ \vdots\\ Y_n \end{pmatrix}
 = \begin{pmatrix} h_1(X_{11}) & h_2(X_{11}) & \dots & h_p(X_{11}) & \dots & h_1(X_{1p}) & h_2(X_{1p}) & \dots & h_p(X_{1p})\\ \vdots &
 \vdots  & & \vdots & & \vdots & \vdots & & \vdots\\ h_1(X_{n1}) &
 h_2(X_{n1}) & \dots & h_p(X_{n1}) & \dots & h_1(X_{np}) & h_2(X_{np})
 & \dots & h_p(X_{np}) \end{pmatrix} \begin{pmatrix} \beta_1\\ \vdots\\
 \beta_{p^2} \end{pmatrix}
 + \begin{pmatrix} \epsilon_1\\ \vdots\\ \epsilon_n \end{pmatrix}? \end{align*}

I appreciate any help.

You have it exactly right.
For instance, you might have two predictors $X_1$ and $X_2$. In your model, you decide to use $X_1$ untransformed, "as-is": $h_1(X_1)=X_1$. For your second predictor, you decide to use $X_2$ both untransformed, $h_1(X_2)=X_2$ and squared, $h_2(X_2)=X_2^2$. Your model contains three predictors $X_1, X_2, X_2^2$, so your parameter vector is also of length three.
Unless you also add an intercept, that is. Which in this framework you could consider yet another function, namely the one that sends everything to $1$: $h_0(X)=1$.
