I ask this question because according to Dimensional Anchor's text, it blocks all teleportation based things from happening, at least that's what it is by RAI. However by RAW it blocks only spells and spell-like abilities, which Dimensional Slide is not, because it's a supernatural ability.
Is my understanding correct? Can arcanists slide through dimensional anchors?
No. The spell was written before such abilities existed.
While it isn't "RAW" it's definitely against the rules as intended.
I ask this question because according to Dimensional Anchor's text, it blocks all teleportation based things from happening, at least that's what it is by RAI. However by RAW it blocks only spells and spell-like abilities, which Dimensional Slide is not, because it's a supernatural ability.
Is my understanding correct? Can arcanists slide through dimensional anchors?
By RAW, you could, because the spell calls out spell-like abilities only.
It's the same reasoning that lets people with Deflect Arrows deflect alchemists' bombs:
Deflect Arrows works on ranged weapons that are not spells or spell like abilities. No supernatural effects that function as ranged weapons existed at the time the feat was written, so the feat makes no mention of it. When alchemists' bombs created a supernatural effect that functioned as a ranged weapon, the general consensus was that Deflect Arrows did work on them because the feat references spell-like abilities only. (This was also widely interpreted as the actual intent rather than a RAW loophole: the fact that supernatural effects had not been envisioned at all was considered irrelevant.)
Applying the same logic, the Dimensional Anchor only works on spell-like abilities, so it does not work on Dimensional Slide. Likewise, it would not work on any supernatural or extraordinary teleport abilities from monsters. Whether any of these existed when the spell was written is completely irrelevant.