"In this way" refers to knowing the spell via other sources
As you mentioned, the feat reads:
... you must finish a long rest before you can cast it in this way again.
"In this way" refers to the fact that you cannot cast the spell using the feat, but if you are already able to cast the spell without the feat, then you may cast it in that way. The feat itself does not allow you to cast the spell in any other way, though.
You learn the spell charm person as a Wizard, but then you take Magic Initiate to get some Bard spells, and you choose charm person as your 1st level Bard spell (unlikely that someone would do this, but possible). The wording of the feat says you can cast this spell once per long rest as per the feat, but you would still be able to cast it with Wizard spell slots because you already knew it as a Wizard spell.
Another example is that you are a Sorcerer, and you take Magic Initiate to learn more cantrips/spells from the Sorcerer list. Whatever you pick counts as a Sorcerer spell for you because you're already a Sorcerer, so although you can cast the spell once per long rest via Magic Initiate, you may also cast the new spell you gained via the feat using your spell slots.
This is clarified in the Sage Advice Compendium:
If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat?
Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell.
If you have spell slots, can you use them to cast the 1st level spell you learn with the Magic Initiate feat?
Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.
Just to reiterate, Magic Initiate only allows you to cast the spell you choose once per long rest. To cast it again using spell slots, you must have already had the means to do so.