Emphasis mine:

Any creature can cast a spell of 1st through 5th level into the ring by touching the ring as the spell is cast. The spell has no effect, other than to be stored in the ring.

It does not say that any spell cast while touching it must be stored into the ring. Interpreting it like that would mean that a caster wearing this ring would be basically unable to use many of their spells.

This would not only require twisting the RAW wording (by using a non standard definition of "can"), but also feels like against RAI in your example: an item that could be used to repeatedly cancel the casting of any spells under 5th level that only requires a reaction to use and that is never expended is by comparison way stronger than everything else in the rare category. Compare that to the legendary Ring of Spell Turning which merely grants advantage on saving throws and counters spells completely only when you get a natural 20. Or compare with Rod of Absorption which is very rare and has a limited number of charges.

Spells cast from the ring use normal actions and no components.

Many magic items allow the user to cast spells. The general rules for this are on p. 141 of the DMG, under the Spells subsection of Activating an Item:

Some magic items allow the user to cast a spell from the item. The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration.

(emphases mine).

Beyond these rules for magic items, casting a spell follows the rules in the Player's Handbook; generally, the character casting the spell takes the Cast a Spell action on their turn.

Here's what the Ring of Spell Storing says about casting the spells in it (DMG, p. 192:

While wearing this ring, you can cast any spell stored in it. The spell uses the slot level, spell save DC, spell attack bonus, and spellcasting ability of the original caster, but is otherwise treated as if you cast the spell.

Let's look at two examples of stored spells being cast:

1. Fireball stored in the ring at 3rd level by a 10th Level School of Evocation wizard with a save DC of 16
2. Magic Circle stored in the ring at 5th level by a 11th level Cleric with a save DC of 17.

Fireball, cast from the ring by a 3rd level Sorcerer (save DC 14):

• Will require one action to cast (same as the spell)
• Will do 8D6 damage; the original casting wizard's Empowered Evocation feature will not apply since it is not one of the characteristics that the ring lists as being 'stored'.
• Can be used with Quickened Spell by the casting sorcerer, since that falls under being "treated as if you cast the spell".
• Will have a save DC of 16, from the original caster.

Magic Circle, cast from the ring by a 19th level Fighter:

• Will require one minute to cast (same as the spell).
• Will not require any chalk or other material components, even though that is normally required.
• Will have a duration of three hours (base duration plus two extra hours for using a 5th level slot).
• The fighter has no spellcasting ability, and thus no spell save DC, but the Magic Circle will have a save DC of 17.