Reinhardt is a pure-blooded Belmont. The only difference is his sir name. Many make the mistake that Patriarchy is the only way to have a pure bloodline. That's a terrible misconception that Patriarchal belief has established in society. It's garbage actually, if not sexist. If a woman of Belmont blood had a child with someone outside of the family that would not make the child any less of a Belmont. It's no different with the man of Belmont blood having a child with a woman outside of the family. The only difference is the name change. We have to keep in mind as well that CV64 (Apocalypse) came out before IGA had established his own personal CV mythos. Though, no doubt, it was in the works.
there's different ways to look at it though. to those that only acknowledge the absolute pure blooded and maintain pure blood within a family, they would have to resort to incest to maintain that pure blood family tie. Of course any family can start with unrelated pair producing a new line of relatives, however for a family bloodline to remain at its purest instead of mixed, there would have to be internal family reproduction to maintain the bloodline at its more pure.
that's where family clans come in as well, which can have branches of relatives all coming from the same source of purebloods.
with Belmont's case, it's flexible because the Belmont's clan agenda is not to maintain a family of inbred, but pure blooded Belmont is really not the case when from what I read, Belmonts really don't have strict guidelines who a belmont can bond with.
yes, someone from the Belmont line and different last name, is still a Belmont in a sense, but actually they are only completely a Belmont if they themselves perceive themselves as a Belmont. just the family name changing doesn't remove the blood relations they have with both their parents.
Now someone that is totally unrelated, could be adopted by a Belmont family, and be acknowledged as a Belmont, even without blood relation. Just as someone that deserts the Belmont family, and abandons the family name, is not longer a Belmont by name but could be regarded as a Belmont by blood, though the blood relation doesn't matter if the individual cuts ties with that family title. People can change after all, including the title. Some may say they are still related, however, might as well say everyone is related since practically an entire species is from the same source/originals of their kind.
So there are different tiers and spectrums to identify someone's name and relation. really, even family titles are more of a mutual acceptance thing. so it's really unwritten rules, or illusionary rules that a family chooses to follow if they agree with those rules.
if Miriam secretly had bloodties to Shanoa, for example, whether they know it or not, they are blood related, however they are considered unrelated since there's no proof, so even if Shanoa were to have a last name, Miriam wouldn't inherit that name if there's no family bond there to create family ties with a name, stuff like that. So if Miriam is Shanoa's daughter, she would be related in blood only, without a shared family name.
family name really simply expresses family bonds and confirms relations, whether blood related or adopted.