The definition of a 2-chain comes from Fayer's paper at: https://qmro.qmul.ac.uk/xmlui/bitstream/handle/123456789/64468/Fayers%202-chains%3A%20an%20interesting%202020%20Accepted.pdf?sequence=2&isAllowed=y
In the paper he defined 2-chains to bear the following 2 conditions: 1. There is a unique way to write P as the union of 2 chains. 2. $<$ is maximal subject to 1, i.e. if $<^{+}$ is a proper refinement of $<$, then there is more than one way to write P as the union of 2 $<^+$ chains. After this main definition he claimed that there will only be 1 maximal element that is greater than every other non maximal elements in a 2-chain(call it super maximal).
What about the following "2-chain": we have $x_1<x_2<x_3$ on the left and $y_1<y_2$ on the right? Clearly there is no super maximal element. What is this type of poset called?
The best term I know for a poset of the type you describe is simply "disjoint union of two chains." It's certainly not a $2$-chain; we can see this by noting the absence of a unique supermaximal element, or more directly by observing that its order relation is not maximal since we can set $x_3>y_1$ and $y_2>x_2$ while preserving the other conditions.
For completeness, here's the picture from the top of page $2$ of the cited paper showing the four isomorphism types of five-element $2$-chains; note that the extension I describe above is the rightmost example: