State this proposition for an $\mathcal{L}$-embedding using this convention I have started learning logic, and I have been finding it difficult because there seems to by a relatively large disparity between different types of notation in this field of maths compared to other fields  of maths. So far I have come across two different conventions for valuations of variables. I have been reading this text and have so far well-understood it https://www3.nd.edu/~cmnd/programs/cmnd2016/undergrad/Conant_MTnotes.pdf. In this text, the way they assign values to variables, is to treat each variable as an identity function, and then the values we want to assign to those variables are just plugged into that function. The second convention I have seen for valuations is set up like this:
$W^\mathcal{M}=\{w|w\text{ is a function }w:\mathbb{N}\to M\}$. We use $w\in W^\mathcal{M}$ to get such a mapping by declaring that $v_k$ is mapped to $w(k)$.
Let $w\in W^\mathcal{M}$, and let $p\in M$. Then $w(p/i)\in W^\mathcal{M}$ is defined by
$$
w(p/i)(j)=
\begin{cases}
w(j)&\text{if }j\neq i\\
p&\text{if }j=i.
\end{cases}
$$
Thus $w(p/i)$ is a valuation that assigns $p$ to $v_i$ irrespective of what $w$ did, but leaves all other values of $w$ unaltered; it thus differs from $w$ only at the "$i$th place".
We have a proposition about embeddings written using the first convention:
For an $\mathcal{L}$-embedding $\pi$ between structures $\mathcal{M}$ and $\mathcal{N}$, we have that for a variable $v_i$
$$
\pi(v_i^\mathcal{M}(a))=v_i^\mathcal{N}(\pi(a)).
$$
How would I write this proposition using the second convention for valuations?
 A: Let me sum up some of the ideas expressed in the comments. We are considering two different things here – embeddings and valuations – which are fundamentally very different ideas. Let's try to understand what they are and how they differ; fix a first-order language $\mathcal{L}$.
A valuation is a kind of map that involves only a single, fixed $\mathcal{L}$-structure $\mathcal{M}$; at its most basic level, a valuation on $\mathcal{M}$ is just an assignment of variables in $\mathcal{M}$. In other words, the data of a valuation $w$ is precisely a choice of some element $m_k\in M$ for every variable $v_k$ of the language. The utility of valuations is that they allow us to reason about the internal logical structure of $\mathcal{M}$, in the following sense. A first-order formula $\phi$ will generally involve a number of "free variables". For example, the formula $\phi\equiv \exists v_2(v_1\neq v_2)$ has a single free variable – namely, $v_1$ – and can therefore be thought of as expressing a property of the variable $v_1$. (What property is this choice of $\phi$ expressing?) On its own, $\phi$ is not very useful; indeed, if we are considering the logical properties of the structure $\mathcal{M}$, we want to know how the elements of $\mathcal{M}$ behave, and $\phi$ as written does not involve any elements of $\mathcal{M}$ – it has variables instead. The way to fix this is by considering suitable valuations; if we want to know whether the sentence $\exists v_2(m\neq v_2)$ holds for some $m\in M$, the right approach is to consider those valuations $w:\mathbb{N}\to\mathcal{M}$ such that $w(1)=m$, since those valuations are precisely the ones that encode the idea of "substituting $m$ in for the free variables $v_1$". The whole purpose of the machinery and details that you have cited in your post is thus to make precise what it means for a formula to hold when we substitute elements of $\mathcal{M}$ in for its free variables. Once these technical details are dispatched with, we have a framework for turning abstract $\mathcal{L}$-formulas into a tool for thinking about the internal structure of $\mathcal{M}$, and in particular for thinking about how the elements of $\mathcal{M}$ interact with each other.
An embedding is a completely different beast. While valuations are maps involving only a single $\mathcal{L}$-structure $\mathcal{M}$, embeddings are maps involving a pair of $\mathcal{L}$-structures $\mathcal{M}$ and $\mathcal{N}$. If valuations are a tool for understanding the internal structure of a single $\mathcal{L}$-structure (ie, understanding how its elements interact with each other), then embeddings are a tool for understanding the external structure of multiple $\mathcal{L}$-structures – namely, understanding the ways that the $\mathcal{L}$-structures themselves can interact with each other. Of course, we're not just interested in any old functions between $\mathcal{M}$ and $\mathcal{N}$; since we are thinking of $\mathcal{M}$ and $\mathcal{N}$ as $\mathcal{L}$-structures, we also want embeddings to play nicely with the ambient $\mathcal{L}$-structure (the jargon is that we want an embedding to be a "structure-preserving map" $\sigma:\mathcal{M}\to\mathcal{N}$), and this motivates the definition of an embedding that you have described in your post. An embedding $\sigma:\mathcal{M}\to\mathcal{N}$ lets us identify $\mathcal{M}$ with an $\mathcal{L}$-substructure of $\mathcal{N}$ (which one?), and allows us to ask what properties are preserved or negated when moving from $\mathcal{M}$ to $\mathcal{N}$. These kinds of questions are "external" in the sense that they can involve multiple $\mathcal{L}$-structures at a time.
To sum things up, when we are working with valuations, the ambient "universe" that we are working in is a single, fixed $\mathcal{L}$-structure, and the main question we are interested in is the way the elements of that structure interact with each other. When we are working with embeddings, the ambient universe we are working in is the collection of all $\mathcal{L}$-structures, and the main question we are interested in is the way that those $\mathcal{L}$-structures interact with each other. This is a slightly reductionist view of things, but it's at least a useful slogan for understanding why embeddings and valuations are entirely different kinds of objects; it's not just that they are defined differently or employ different notations, but that they have completely different scopes and uses. So, when you write

I really couldn't come up with a good way of expressing the idea of an $\mathcal{L}$-embedding using this notation,

that is not a bad sign, since these two notions involve different ideas! Instead of trying to relate them to each other, it is best to see them as being two distinct tools, which we use in different circumstances depending on what questions we are interested in answering. To use a metaphor, one is like a screwdriver and one is like a hammer; although we could use a screwdriver as a hammer in a pinch, that's not the right way to use it, and it's best to see them as distinct tools with their own distinct uses!

Edit: Following your edit to your post and your comment below, am I correct that you are asking whether there is a way to express that embeddings commute with term evaluation using the notation for valuations in your post? If so, the answer is yes. Fix an embedding $\pi:\mathcal{M}\to\mathcal{N}$. Then, for any valuation $u$ on $\mathcal{M}$, we can define the "pushforward" valuation $\pi\circ u$ on $\mathcal{N}$. If $u$ assigns a variable $v_i$ to an element $m_i\in M$, then $\pi\circ u$ assigns $v_i$ to $\pi(m_i)\in N$. Now, using the notation from your second source, given any term $t$, we will have $\pi(u^*_\mathcal{M}(t))=(\pi\circ u)^*_\mathcal{N}(t)$; as an exercise, prove this by induction on the term $t$. In particular, when $u$ maps the free variables of $t$ to the tuple $\bar{a}$, then this equation expresses precisely that $\pi(t^{\mathcal{M}}(\bar{a}))=t^{\mathcal{N}}(\pi(\bar{a}))$, where the latter equation uses the notation of Gabe's notes. Is that what you were looking for?
A: Hello @Atticus Stonestrom. I'm stuck for doing this induction...
With the old convention it makes sense to me:
Let $t=f(t_1,\dots,t_n)$, and the proposition holds for all $t_i$. Then
$$
\begin{align*}
\pi(t^\mathcal{M}(\bar{a}))&=\pi(f^\mathcal{M}(t_1^\mathcal{M}(\bar a),\dots,t_n^\mathcal{M}(\bar a)))\tag{def. of interpretation of functions}\\
&=f^\mathcal{N}(\pi(t_1^\mathcal{M}(\bar a)),\dots,\pi(t_n^\mathcal{M}(\bar a)))\tag{def. of embedding}\\
&=f^\mathcal{N}(t_1^\mathcal{N}(\pi[\bar a]),\dots,t_n^\mathcal{N}(\pi[\bar a]))\tag{by inductive assumption}\\
&=t^\mathcal{N}(\pi[\bar a]).\tag{def. of interpretation of functions}
\end{align*}
$$
So I tried to follow this with the second convention and got stuck:
$$
\pi(w_\mathcal{M}^*(t))=\pi(f^\mathcal{M}(w_\mathcal{M}^*(t_1),\dots,w_\mathcal{M}^*(t_n)))\tag{def. of valuation of functions}
$$
and then I got stuck trying to apply the definition of embedding onto this quantity. Sorry I know these are really really basic questions :/
