The concept of "correct" isn't really appropriate here. Semantically, both sequences are identical (there is no difference; it's just as much a difference without a distinction as a distinction without difference). And here's the evidence that difference without a distinction has been written many many times - it's just less common.
There is also the phrase "a distinction without a difference". This means that that it is possible to distinguish two things, but that the difference between them does not matter in any way that is important (to the author). In addition to the relevant definition ಠ-ಠ provides, "differentiate" has a mathematical meaning.
¡Hola a todos/as! Estoy traduciendo un contrato y me ha aparecido esta expresión: "This is a distinction without a difference". Entiendo lo que quiere decir, pero no sé cómo traducirla, a ver si podéis ayudarme. ¡Muchas gracias de antemano! Aquí se explica la expresión...
I had a professor who would jokingly say, "Is this a difference without distinction or a distinction without a difference?" Anyway that doesn't help. Difference is more fundamental, or can be, and certainly is used more often. Distinction has to do with a defining characteristic. So as Cagey says, put some words in the blank and it would be much easier to discuss.
With transitive provide sth to/for sb, I think answer 2 is closer - to is more about giving or handing off something to someone, while for is more about something being made available to someone. Most of the time it is a distinction without difference, and there's no solid line dividing the two, but there is a bit of a nuance: A laptop was provided to me. (more likely that someone came to me ...