How should I interpret "two discs intersect along an arc" and "union of discs" from Armstrong's Basic Topology?

Question

In Armstrong's Basic Topology, there is a lemma stated as follows:

Let $A$ and $B$ be be discs which intersect along their boundaries in an arc. Then $A\cup B$ is a disc.

I am not sure what the author actually means. I have proposed two interpretations:

$1.$ We start with a topological space $X$ and two subspaces $A$, $B$ of $X$. Suppose each of $A$ and $B$ is a disc, and $X=A\cup B$, and the intersection $A\cap B$ is on the boundaries of each of $A$ and $B$, and $A\cap B$ is homeomorphic to $[0,1]$. Then $X$ is a disc.

$2.$ We start with two discs $A$ and $B$ and a continuous function $f$ which embeds an arc of $B$ (which is a subset of the boundary homeomorphic to $[0,1]$) to the boundary of $A$. Then the adjunction space $A\cup_f B$ is a disc.

Are any of these interpretations correct? Or are there some better interpretations?