Does Mystic Vortex only include cards that have the Reaction keyword on it and not include cards that play on Reaction speed like Ambush and Quickdraw cards?

No, Mystic Vortex doesn't only apply to cards with the printed Reaction keyword — it applies to any card that has Reaction as a characteristic at the time of play, regardless of how it got it.

Here's why:

Quick-Draw cards are affected. Rule 819.1.b says "Cards with Quick-Draw have Reaction inherently" — they permanently have Reaction as a characteristic, even if it's not a printed keyword on its own.

Ambush cards are also affected, but only when the Ambush condition is met. Rule 822.1.b grants them Reaction "as long as I'm being played to a battlefield where you control Units." If that condition is true during a showdown at Mystic Vortex, the card has Reaction and the cost increase applies (Rule 813.4–813.5).

Hidden cards are explicitly called out in Mystic Vortex's own reminder text: "(Hidden cards have [Reaction].)" This matches Rule 811.6, which says Hidden cards gain Reaction while played from facedown.

The bottom line: Mystic Vortex checks whether the card has the Reaction characteristic (Rule 813.5), not whether the word "[Reaction]" is physically printed on it. Quick-Draw and Ambush cards qualify when those effects make Reaction true for them.