Innate spells only ignore the components that are explicitly listed

Innate spells follow the normal rules for spell components unless otherwise noted

An innate spell can have special rules or restrictions. For example, a drow mage can innately cast the levitate spell, but the spell has a “self only” restriction, which means that the spell affects only the drow mage. (MM 10)

Since innate spells are simply normal spells¹, they follow normal spellcasting rules unless there is a specific rule that overrides the general. This means that innate spells need all the components as normal spells unless a specific rule for that monster says otherwise.

Jeremy Crawford has confirmed this via Twitter (several times):

Q: Does innate spell casting use V and S components?

A: A creature's Innate Spellcasting trait tells you if it doesn't require certain components. It otherwise requires components.

Only innate spellcasting that ignores all components that are required by the spell is not able to be counterspelled

This does not conflict at all with the passage you quote:

If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible.

All this is saying that the trait, innate spellcasting, is possessed by many creatures and is an example of a trait that can potentially make a spell imperceptible. It never implies innate spellcasting automatically makes a spell imperceptible by removing the need for any components. As shown above, that is not at all how the rules say it works.

This is even more easily seen by the fact that it lists subtle spell as well. Since subtle spell explicitly only removes any somatic or verbal components it certainly is not an example of an ability that automatically makes a spell imperceptible. If a spell were to have material components it would still be perceptible.

Thus, only if an innate spell specifically says that no components need to be used for the spell, is that spell unable to be counterspelled.

Examples

Many (if not all) innate spellcasting traits specify that the spells do not require material components. This is specified in the monster statblocks:

The mephit can innately cast dancing lights, requiring no material components. Its innate spellcasting ability is Charisma.

So this spell needs no material components since the ability specifically overrides the general rule. However, since the ability says nothing about verbal or somatic components (which dancing lights requires), those are still required as per the general spellcasting rules.

However the Githyanki Warrior's ability is different:

The githyanki’s innate spellcasting ability is Intelligence. It can innately cast the following spells, requiring no components...

Since it says no components are required, these spells would be imperceptible and thus not counterspellable.

^{1 - The Monster Manual describes innate spellcasting as a monster ability with which they can cast spells:}

^{A monster with the innate ability to cast spells has the Innate Spellcasting special trait.}

^{Jeremy Crawford also explicitly supports this:}

^{A spell is a spell no matter how you cast it: spellcasting, innate spellcasting, racial trait, magic item, class feature, etc.}