Only if they use the ability to attack.
As you've noticed, mystic abilities don't count as spells. As such, they're not subject to the same restrictions as spells. Just like a monk can use Step of the Wind and a druid can Wildshape without breaking invisibility, so can a mystic manifest a power.
However, some mystic abilities involve making an attack roll. This involves a certain amount of interpretation, because "an attack" in D&D terms can mean either the attack action (a specific action that neither casting a spell or manifesting a mystic power counts as) or making an attack roll - ie a d20 roll with an attack bonus against a creature's AC.
The base rules define an attack thusly:
If there’s ever any question whether something you’re doing counts as an Attack, the rule is simple: if you’re Making an Attack roll, you’re Making an Attack.
So using a mystic power that triggers an attack roll is making an attack, and would end invisibility.