It's deeper than No More Heroes. The special attack I mention is something different from the strong attack I mention at an earlier point in the review. With Y being the normal attack and X being the strong attack, the possible combos for most characters are:
So you can either mash the Y button or conclude the combo at various points with the X button. Each of those combos (2-6) will end with a different attack, so not everything works well for every situation.
Regarding finishing moves, there's something I forgot to mention in the review. When enemies perform a strong attack, they expose their weak point. If you deplete that circle over their head, then your character will automatically perform a special combo that hits the enemy hard. It will not necessarily kill them, but it's the most effective way to deplete their health bar.
It's worth noting, in case anybody didn't realise, that some of the moves do let you do more than that. For instance, when using Link with his sword/shield, if you do YYYX, Link will leap into the air with a simple spinning attack... and then you can use Y a few more times followed by X to slash from the air and then stab down with your sword to finish (I actually use the Zelda-inspired controls, so it's BBBY rather than YYYX for me).
Other attacks have other variations - Lana with the Spear, if you use YYYYYYY, will end up shooting out water that turns into ice stalagmites... and then you can either use X or Y to finish. With Y, she blasts the ice, doing damage to those around them. With X, she turns them into Deku Tree Sprouts, which jump around the area doing damage.
Lana with the Book can do a variety of things with those blocks. For instance, YYYXX will summon lightning and then produce three magical walls. Pressing X again immediately will cause them to start moving forwards. However, if you wait a moment, they remain stationary, and pressing X makes her leap to them, after which she will jump into the air and destroy them, damaging those around her (indeed, any of the white magical walls can be destroyed in this way).
Also, regarding weak points, there's one other thing that might be helpful to some people - when it comes to Stalmasters and Darknuts, if you attack them, then dodge mostly-towards them, you'll dodge around them, and they'll reveal their weak spot immediately (assuming they don't start an attack at the same time). Makes it a lot easier to take them out, especially when you're in a rush.