You prove someone guilty by proving the presence of every element of an offense beyond a reasonable doubt. "Innocent until proven guilty" means innocent until every element of an offense is proven beyond a reasonable doubt.
This is, of course, a rebuttable presumption. That is, that while the legal system should presume that someone is innocent of all elements of all offenses, that presumption can be rebutted by demonstrating facts and evidence to suggest that the truth is otherwise. This is a presumption in favor of the defense that the onus is on the prosecution to rebut to obtain a conviction.
Note that just requiring some elements to be established this way does not suffice. For example, if we were to imagine some jurisdiction where rape had elements of both sex and lack of consent but the prosecution only had to prove that intercourse occurred to get a conviction, that would violate the concept of innocent until proven guilty.
People have sex all the time without committing rape. If all the prosecution had to prove was that someone had sex and then the defense had to prove consent, that would be an absence of "innocent until proven guilty". The presumption of the presence of an element of an offense is a form of "guilty until proven innocent".
A person who has sex has not committed the crime of rape. But if we presume they did unless they can prove consent, then they are guilty until proven innocent.
But there are any number of presumptions in favor of the prosecution that exist in various legal systems. In this case, even if the presumption is required for the establishment of guilt, the burden is still on the accused to rebut the presumption.
Some systems even have irrebuttable presumptions. If an irrebuttable presumption is necessary to establish someone's guilt, the person is presumed guilty, period. They cannot even present evidence to rebut the presumption.
For example, in Canada, there is an irrebuttable presumption that one's blood alcohol level drops at more than 5mg/100mL per 30 minutes. If a blood alcohol test done an hour later shows your blood alcohol level to be 8mg/100mL below the legal limit, the presumption suffices to establish your guilt and you cannot even introduce evidence to the contrary such as an earlier test that could prove that your blood alcohol level was below the legal limit when you were actually driving.
That is, you can be innocent of the crime of driving under the influence and even able to prove that your blood alcohol level was below the limit at all the times you were driving, but nevertheless, you can be presumed guilty if the prosecution can show that your blood alcohol level was high, but still below the legal limit soon after you were driving. This is so even if you have conclusive proof your blood alcohol level was never over the legal limit while you were driving. The prosecution is entitled to an irrebuttable presumption of guilt in this case.
UK rape law has some conclusive presumptions as well. For example, if you are accused of raping someone and the prosecution can show that you intentionally impersonated someone known to the complainant, there is an irrebuttable presumption in favor of the prosecution that there was no consent.
In other words, in the UK, if you have sex with someone after impersonating someone they know, you are presumed guilty of rape because there is a conclusive presumption that they did not consent. You cannot even present evidence that they consented -- the law requires the jury to presume that they did not consent even though the absence of consent is a required element of the crime of rape.
This is "guilty even if proven innocent". No matter how much evidence of actual consent you present -- say even a video recording of the "victim" consenting -- you are still guilty of rape and the jury must, by law, conclude that the victim did not consent.
Rebuttable presumptions are more like "guilty until proven innocent" though. Here, the Prosecution does not have to provide evidence to establish some fact necessary for guilt, instead the defense has to provide evidence to show the presumed fact is not so.
For example, if you have sex with someone who you are unlawfully detaining, UK law entitles the prosecution to a presumption that the sex was non-consensual. Non-consent is an element of rape, but the prosecution need not prove it in this case. It is on the defense to prove the absence of consent.
Australia has a "reverse onus" jury instruction for cases involving rebuttable presumptions. It is something like, "In this case, there is an exception to the general rule that the prosecution must prove their case beyond a reasonable doubt. The law says that for an offence of [whatever], the accused must prove [whatever] rather than the prosecution."
When the thing the defendant must rebut is an element of the offense, then there is no innocence until proven guilty. Even though the prosecution has not proven an element of the offense, the defendant may be presumed guilty until and unless they can prove the absence of that element.
For example, say you are an Australian charged with unlawfully entering an area that the government of Australia has declared may not be entered. That you did not enter solely to provide humanitarian aid is an element of the offense. However, you have the burden of demonstrating that you only entered to provide humanitarian aid. It is not even enough to show that you did in fact provide humanitarian aid while there, you must somehow establish that this was your sole purpose. The jury will get a "reverse onus" instruction that you must somehow prove that your sole purpose was to provide humanitarian aid.
In other words, here is a crime that has as an element that you intended to do something other than render humanitarian aid. But the prosecution does not have to prove this -- you do. This is a case where the defense must prove the absence of something (best of luck doing that) or they are presumed guilty of this element of the offense. If this is the only absent element, then they are presumed guilty of the crime.
What I mean by this is that there are non-US jurisdictions that allow defendants to be convicted on evidence that would be thrown out of a US court as unconstitutionally obtained, but the prosecution still has to present this evidence and use it to convince the court of the defendant's guilt. The court does not simply assume, "The prosecutor said it so it must be true, GUILTY!".
I think this part of the question is based on a misunderstanding. If evidence is unconstitutionally obtained, the harm is completed when the evidence is obtained. The introduction of reliable evidence in a criminal trial, even if wrongfully obtained, does not violate any right recognized by US law.
The right not to have the evidence wrongfully obtained has already been violated, but no US court has ever held that you are entitled to have the evidence suppressed as a remedy for that violation if there is no reason to think the evidence is unreliable. United States law does not recognize any right to have wrongfully-obtained evidence excluded except where the way the evidence was wrongfully-obtained would tend to make that evidence inaccurate. For example, a confession obtained through torture.
Otherwise, wrongfully obtained evidence can be excluded where a court feels the benefit of excluding the evidence exceeds the harm from excluding it. But this is not done because you have any right to have the evidence excluded, it's done because it makes sense to do things whose benefits outweigh their harms. The harm would be that a jury in a criminal case can't access some relevant facts. The benefit is that wrongdoing may be discouraged in subsequent cases.
More often than not, courts find that the benefit does not outweigh the harm and they admit wrongfully-obtained evidence. For example, if the police violated someone's rights in obtaining the evidence but had a good faith belief that what they were doing was lawful, what's the benefit of excluding the evidence? You can't really discourage accidental wrongdoing that way, or at least that's what the Supreme Court held in United States v. Leon.
Suppression of evidence, however, has always been our last resort, not our first impulse. The exclusionary rule generates “substantial social costs,” United States v. Leon, 468 U. S. 897, 907 (1984) , which sometimes include setting the guilty free and the dangerous at large. We have therefore been “cautio[us] against expanding” it, Colorado v. Connelly, 479 U. S. 157, 166 (1986) , and “have repeatedly emphasized that the rule’s ‘costly toll’ upon truth-seeking and law enforcement objectives presents a high obstacle for those urging [its] application,” Pennsylvania Bd. of Probation and Parole v. Scott, 524 U. S. 357, 364–365 (1998) (citation omitted). We have rejected “[i]ndiscriminate application” of the rule, Leon, supra, at 908, and have held it to be applicable only “where its remedial objectives are thought most efficaciously served,” United States v. Calandra, 414 U. S. 338, 348 (1974) —that is, “where its deterrence benefits outweigh its ‘substantial social costs,’ ” Scott, supra, at 363 (quoting Leon, supra, at 907). -- Hudson v. Michigan