In order to define a free product with amalgamation $A{\ast }_{C}B$ you need to define $A,B,C$ and embeddings of $C$ into $A$ and $B$. Alternatively, instead of $C$ you can use a homomorphism $phi:A\to B$ and its inverse. A third option is to pick subgroups $D\subset A$ and $E\subset B$ and give an isomorphism between them.

All three of these options require you to define some groups and some morphisms. You can define a group nicely by giving a presentation but defining maps usually takes a few lines. So, I’d like a quicker way of describing the kinds of maps we need here.

Let’s take the third option. If we have generating sets so that $D=⟨X⟩$ and $E=⟨Y⟩$, then if we can write each of the elements of $X$ as words in $Y^{\ast }$ then that defines the required isomorphism completely. After you’ve done that, the generating set $Y$ might as well just be those words.

So how about this as a presentation for an FPA?

$$⟨a_1,a_2,\dots |R_A⟩[x_1=y_1,x_2=y_2,\dots ]⟨b_1,b_2,\dots |R_B⟩$$

The bits in angle brackets are presentations of $A$ and $B$. The $x_i$ are words from $A$ and generate $D$, and the $y_i$ are words from $B$ and generate $E$.

Because this is all on one line and delimited adequately, you can chain things together and use brackets and all that, like this contrived example:

$$(⟨a,b⟩[b=c]⟨c,d|cd=dc⟩)[a=e]⟨e,f⟩$$

For HNN extensions, you just need to say what the original group is, what the extending letter is, and what the identified subgroups are. So how about something like:

$$⟨g_1,g_2,\dots |R_G⟩[t|x_1=y_1,x_2=y_2,\dots ]$$

Apart from the different kinds of brackets being hard to distinguish visually, what have I missed?