There are, of course, an incredible amount of theories on how Harry Potter survived. Problem is that the most important one, the one that is canon, is the one I don't get: how Harry's blood inside Voldemort saves him?

Because all that I have read so far, leaves me with questions. I understand why Voldemort wanted Harry's blood, but why does it save Harry? Why is that necessary to keep the sacrifice alive, while it's still inside Harry? And if Harry lost it at 17 (I have seen some different ideas on this as well), why does Voldemort still have it at 69 (give or take a few)?

To clafiry, the parts I'm talking about:

"For a fleeting instant, Harry thought he saw a gleam of something like triumph in Dumbledore's eyes"

(GoF, right after Harry tells Dumbledore that his blood now goes through Voldemort.)

And:

"He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you"

(Albus Dumbledore in DH)

I just don't get why the blood transfusion thing suddenly gave Harry the chance to survive the whole thing, and it's never really been satisfyingly answered, not to my knowledge at least, as every answer I've read raises more questions.

The Dark Lord kept Harry alive by keeping Lily’s sacrifice alive.

The Dark Lord having Harry’s blood inside him meant that Harry’s blood (and therefore Lily’s sacrifice) was kept alive in a body that wasn’t Harry’s own. That tied him to life somewhat like a Horcrux would, except instead of his soul being kept in an external object, it was his mother’s sacrifice inside the Dark Lord. The benefit to the sacrifice in Harry’s blood being in the Dark Lord was that it was kept alive by someone else as well as being in Harry, acting as an anchor for Harry. It tied Harry to life similarly to how having Horcruxes kept the Dark Lord alive when the Killing Curse rebounded and would have killed him if he hadn’t been externally tethered to life.

“He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you and so does Voldemort’s one last hope for himself.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

J.K. Rowling stated this on the F.A.Q. on her old website - the blood in the Dark Lord acted like a Horcrux for Harry and the blood in him kept the sacrifice (and therefore Harry) alive while he was.

Having taken Harry’s blood into himself, Voldemort is keeping alive Lily’s protective power over Harry. So Voldemort himself acts almost like a Horcrux for Harry – except that the power of Lily’s sacrifice is a positive force that not only continues to tether Harry to life, but gives Voldemort himself one last chance (Dumbledore refers to this last hope in chapter 35). Voldemort has unwittingly put a few drops of goodness back inside himself; if he had repented, he could have been healed more deeply than anyone would have supposed. But, of course, he refused to feel remorse.
_{^{- FAQ on J.K. Rowling’s website}}

It was different than it being just inside Harry, since the Dark Lord kept the sacrifice alive in him as well, even if Harry was hurt to the point of dying - that tethered Harry to life like a Horcrux would.

In addition, the protection that Harry’s mother gave him is different than the charm on Number 4 Privet Drive. The protection in his blood was automatically given when Lily died for him, and that was what the Dark Lord kept alive in his body. The shield on Number 4 Privet Drive was put there by Dumbledore after Lily’s sacrifice. It was location-specific (it only protected him in Privet Drive itself) and automatically broke when Harry turned 17. The Dark Lord didn’t keep that one alive.

TL;DR; Lily's sacrifice protected Harry from being killed by Voldemort and became a part of him physically. Because some of Harry's blood still survived (in Voldemort) it protected Harry from Voldemort again in the Forest during the Battle of Hogwarts.

"Your mother died to save you. If there is one thing Voldemort cannot
understand, it is love. He didn't realize that love as powerful as
your mother's for you leaves its own mark. Not a scar, no visible
sign...to have been loved so deeply, even though the person who loved
us is gone, will give us some protection forever. It is in your very
skin." - Dumbledore

--Harry Potter and the Philosopher's Stone

The protective charm on Privet Drive was something different and was placed there by Dumbledore. That is what would break when Harry turned 17:

"[Petunia] may have taken you grudgingly ... yet still she took you,
and in doing so, she sealed the charm I placed upon you. Your mother's
sacrifice made the bond of blood the strongest shield I could give
you."

"I still don't --"

"While you can still call home the place where your mother's blood
dwells, there you cannot be touched or harmed by Voldemort. ... You
need return there only once a year, but as long as you can still call
it home, there he cannot hurt you. ..."

--Harry Potter and the Order of the Phoenix

In Goblet of Fire, Voldemort took Harry's blood.

"He said my blood would make him stronger than if he's used anybody
else's" Harry told Dumbledore. "He said that the protection my- my
mother left in me - he'd have it too. And he was right - he could
touch me without hurting himself, he touched my face".

For a fleeting instant, Harry thought he saw a gleam of something like
triumph in Dumbledore's eyes. But next second Harry was sure he had
imagined it, for when Dumbledore had returned to his seat behind the
desk, he looked as old and weary as Harry had ever seen him.

"Very well", he said, sitting down again. "Voldemort has overcome that
particular barrier."

-- Harry Potter and the Goblet of Fire - Chapter Thirty Six - The Parting of the Ways.

And in Kings Cross:

‘He took my blood,’ said Harry.

‘Precisely!’ said Dumbledore. ‘He took your blood and rebuilt his
living body with it! Your blood in his veins, Harry, Lily’s protection
inside both of you! He tethered you to life while he lives!’

--Harry Potter and the Deathly Hallows

It changed the protection from direct curses and contact from Voldemort, to now a tether to life protecting from death by curses from anyone including Voldemort, at the cost of protection from direct minor curses and contact.
Though the end result would be the same.

Had Voldemort used another's blood, Harry would still not have died in the forest with Voldemort's soul fragment that was like a horcrux because the elder wand would not kill it's master. The curse would still have rebounded again in the great hall because the elder wand would not kill it's master.

Had Voldemort also used another's wand/had a new wand made/became the master of the elder wand, then Harry would still not have died in the forest because of Lily's sacrificial protection. The curse would still have rebounded again because of Lily's sacrificial protection.