Skill chaining is a special effect that causes fluid movement between two skills. A notable example of this is Mercedes' Charge Drive -> Rolling Moonsault -> Gust Drive -> Lightning Edge, and Phantom's Brise Carte -> Phantom Charge -> Twilight. There is a set timing window in where the next skill can be activated to be chained. If this window is missed, there is generally a significantly longer delay in using the next skill. The skill being chained generally requires activation during the current skill, not after, and thus activates immediately. This separates it from just consecutive spamming of two different skills. Along with this, most skill chains have a special property called "skipActionFrame".
In the data, "skipActionFrame x" designates that when this specific skill successfully chains to another, frame "x" and afterwards are skipped, reducing the delay of the skill. i.e. the skill ends sooner than it normally would, making it faster. Some skills also have a specialActiomFrame/delay, but I do not believe this actually affects the attack speed, as Mercedes' skills have them and I have not noticed any variation.
In summary, most chained skills will reduce the delay, making the skills execute faster than if they were to be used on their own.
This information does not account for possible spam delays and terrain effects or character positions that would otherwise prevent their use. i.e. I do not know if there are any restrictions that can occur that prevent these chains from being executed. This is merely what is presented in the data.
Of Hayato's skills, only the first job skills have prerequisites for activation; that is, you are required to use [Sanrenzan] and [Renjinzan]'s first hit before the second, etc, but no other skills require anything to activate. Any different instances of [Sanrenzan]'s icon is purely stylistic. Any level of the skill itself is interchangeable.
First JobSecond JobThird Job4th Job
Fiel for data
Locked for his opinion and skill icons and being a weiner.