Casting a spell uses the 'Cast a Spell' action.
From the section 'Actions in Combat':
When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise.
Cast a Spell
Spellcasters such as wizards and clerics, as well as many monsters, have access to spells and can use them to great effect in combat. Each spell has a casting time, which specifies whether the caster must use an action, a reaction, minutes, or even hours to cast the spell. Casting a spell is, therefore, not necessarily an action. Most spells do have a casting time of 1 action, so a spellcaster often uses his or her action in combat to cast such a spell.
You are casting a spell only when you take the Cast a Spell action. The actions described in those particular spell descriptions are not the Cast a Spell action, rather they are "an action you gained form a special feature".
Counterspell does not work here.
Counterspell's trigger says:
which you take when you see a creature within 60 feet of you casting a spell.
Since the actions mentioned in these spell descriptions are not "casting a spell", counterspell cannot be used.
Mage Slayer does not work here.
Similar reasoning as counterspell. The first point of Mage Slayer says:
When a creature within 5 feet of you casts a spell