You only need to buy it at the airport if you want a Welcome Suica. You can buy a regular Suica at any station with a multi-function machine.
The only real differences between the Welcome Suica and a regular one are the 500 yen deposit (Welcome Suica doesn't require a deposit) and the limited validity (Welcome Suica automatically expires after about a month).
These sequences are also known as "cubefree," so you might want to continue researching along those lines.
In particular, the game discussed is trying to find cubefree words over a two-letter alphabet. The sample infinite game seems to agree with the listed sequence on OEIS for the lexicographically earliest infinite cubefree word, though your method of generation appears to be different from the one in the comments. (I haven't analyzed it in detail.)
The "positive" surface already contains all the necessary points. It's hard to prove that this surface on its own intersects with itself, but turning it into a Klein bottle makes the proof easy, since it's already known that the Klein bottle must intersect with itself when embedded in 3-D space.
It takes some rigor to ensure that mirroring the surface and turning it into a Klein bottle doesn't introduce a problem that would invalidate the proof, but the idea is this:
1) The surface exists only in the "positive" area above the x-y plane, and the mirror exists only in the "negative" area below the x-y plane.
2) The two surfaces only share the points on the original curve (on the x-y plane), and these points correspond only to the trivial cases where A=B. The surface and its mirror don't intersect anywhere else.
3) The resulting combined surface is a Klein bottle in 3-D space, which must intersect somewhere. Because of 2), that intersection must either be in the positive space or the negative space. Either way, that means there is an intersection in the original surface.
As briefly mentioned in the video, it's critical that the original constructed surface is only in the positive area, because otherwise when you mirror it and then turn it into a Klein bottle, the required intersection might just be the surface intersecting with the mirror, and not within the original surface itself.
"Confession" is fine as a translation. 告白 can be used for admitting to crimes, confessing one's love, revealing a secret, etc. It would never be used for a declaration where there is no sense of guilt/shame/embarrassment involved.
The only real differences between the Welcome Suica and a regular one are the 500 yen deposit (Welcome Suica doesn't require a deposit) and the limited validity (Welcome Suica automatically expires after about a month).