In the case of Scots and Scottish English, there's a lot of blurring and the distinction is of questionable actual value -- though I would make that distinction personally. I speak Scottish Standard English, and have a parent whose first language was a variety of Modern Scots. I find it very hard to follow them when they've switched into it to speak with someone from their hometown. Due to the difficulty in parsing it, both in my case and in the case of the other parent after 40-ish years of marriage, I'd consider it more than just a dialect.

Both Scottish English and Scots descend from the same root Anglic, but there's been divergence since then -- in the same way Scottish Gaelic and Irish Gaelic are descended from the same root, are somewhat mutually intelligible, but are treated as two different languages. It's useful to make that distinction simply because there is a degree of unintelligibility.

Everyone on this website is speaking an Indo-European language, and I'd bet a majority of them have an Indo-European language as their first language. They're all from the same root, so are they all speaking dialects of Indo-European? Or are they speaking distinct languages? Where, exactly, you decide "this was a dialect, but now it's a language!" I can't say, but that point surely exists. It's obviously muddied by politics and national identity and a hundred other factors, but at some point, surely, a language ceases to be a dialect and becomes a language of its own.