It specifically does say that each slain vampire's soul adds to the power of Dracula (not Mathias) so it implies to me he took the name long before his war on humanity.
Which I'm also not disputing. Dracula had a wife and named his son with "Tepes". Mathias obviously named himself "Dracula" before his war on mankind. Perhaps even centuries before his war on mankind.
The only difference between Lord of the Vampires and Dark Lord imo is that post-Lisa's death (CVIII) he wages his war on humanity, that's the essential difference. Everything else is white noise.
There is a very big difference between both, and being the Demon King is FAR from just "a vampire that wages war on mankind".
On the case of calling himself "Lord of Vampires, King of the Night", it is Mathias who calls himself like that. It's a title. If he has the power to back it up, it still doesn't mean he's the Demon King. It only means he's a very powerful vampire who has named himself the Lord of Vampires.
Now, being the Demon King/Dark Lord is a completelly different beast. It's a position atop the demonic hierarchy granted to the one who possesses the power of Dominance - as seen in Aria of Sorrow and Dawn of Sorrow. It's a power to rule over ALL demons, not only vampires, by enslaving their souls using said Dominance. As Soma states in Aria, it's "the power to rule", which sends Graham in a flying panic. Having a connection to Chaos is also important, though I don't know to what extent exactly.
If one wants or not to wage war on mankind with this power, it's irrelevant. What makes the Demon King is his will to rule over demons and become the opposite of good - the entire plot which the Sorrow games revolves around. Dmitrii, also, arrives at the same conclusion after learning that one doesn't need to be Dracula to be the Dark Lord, he only needs the Dominance, and to gain control over demons with it.
The japanese term (魔王 Maō) used to name Dracula is the same term for "Satan", btw, which caused some mishaps as translating the "Dark Lord Ring" as "Satan's Ring" and causing people to believe there actually is a Satan to whom Dracula responds to, when in fact the ring belongs to Dracula, and he responds to no one.
The manual for The Adventure also makes a clear distinction between Dracula as a powerful vampire sorcerer and as a Demon King.
On a side note if Dracula's soul is absorbed into the CS over and over, it wouldn't be Death doing it for him, in nearly every CV game Death is defeated shortly prior to Dracula. Dracula would be absorbing his own soul over and over again.
I don't know where did you get that Dracula's soul is absorbed into the Crimson Stone over and over, but IGA has already said when Dracula is defeated, his soul is sent to the Makai (Demon Realm).