Time PK and Dimension PK are non-canon from each other; they have no connection other than dimension summoning.

This explains the diverging forms of PK: Time and Dimension.

The spark of the divergent starts when PK decides to learn how to train time from Sakuya.

In the universe where PK becomes time, he successfully learns time from Sakuya became the "Judgement of Time".

However, in the universe where PK gains his dimension abilities, he is unable to train with Sakuya due a draft listing, forcing him to participate in the Great Hakurei Barrier defending. He holds off the Barrier from a laser beam, causing him to be the "Synchronizer of Dimensions". After completing his mission of finding the culprit who commanded the attack, he decided that this form was better than time.

I would go and say that Time PK is original Touhou PK while Dimension PK lives in a different dimension. Here's why:

Time PK only uses his time powers whenever he is going against someone else using time or time-related abilities.

With Dimension PK, he permanently uses this form as his main and uses his abilities all the time. This means that the future of PK is drastically different than time.

Actually that doesn't explain a lot but just accept my decisions XD

Now I'm gonna explain the legendary entities: Chronos and Kairos.

Chronos is the entity of time. He controls the flow of time, forwards and backwards.

Time PK's core contains Chronos. Because he found about the legendary entity, his Limit BREAKER spell evolved.

Instead of going berserk, Chronos controls his synergy core, possessing PK as a result. This eliminates the need for a time limit or a risk of death, unless the entity leaves his body, which resumes the effects.

Kairos on the other hand, is the entity of space. He controls all the spacebound dimensions, whether it's 2-dimensional or 3-dimensional. PK's story is different in this era. He met his entity first before having the Berserk DRIVER spell, eliminating that spell all together.

Like Chronos, Kairos can control PK's synergy core, causing him to be possessed. But, unlike the effects if Chronos leaves the host's body during the spell, if Kairos leaves his body, PK will immediately be warped to a dimension where nothing exists.

Basically, PK needs Kairos in order to stay alive when he's using the spell.

Chronos and Kairos have their own spell card where they can be physically summoned out of the user's body and become beings:

Chrono Sign "Epoch Break"

Kairo Sign "Spatial Break"

Chronos and Kairos were actually one single being in the past, but separated due to conflicting ideals.

They can go into their original form, but only for a limited time.

That form, is Caielus.

Caielus the legendary being of  time and space, creating the two elements himelf. Caielus does indeed have spells that are only accessed to him:

Future Sign "Destiny Embarkment": Creates attacks that will unknowingly hit the opponent in the sometime future.

Future Sign "Ascending Hold": Prevents the player from doing a certain action in the future. Uses this spell A LOT.

Sky Sign "Revolving Starlights": Shoots out laser danmaku from the sky that encloses the opponent.  

Chrono Sign "Nega-Tachyon Raging Spark": The strongest Master Spark ever created. Caielus shoots a master spark faster than light time can't follow the beam.

Chrono Sign "Dimensional Tera-Break": A stronger version of Dimensional Outbreak.

Kairo Sign "Transcending Environment Warp": The whole environment becomes disorientating; only Caielus himself can manage this kind of advanced environment

Caielus doesn't have a Limit or BREAKER spell, nor an EXTREMIX spell.

Instead, he has the ASTRIX spell, a category of spells only divine beings can obtain.

ASTRIX Spell

"CAIELUS ~ DIVINE ELEMENTS": Caielus activates the divine powers of time and space. It's godly, no one can resist.

That's all I got, for now.