There's an immediate answer and then there is a deeper nuanced answer which mostly end up in the same place.
No, you cannot tell the truth with a false statement. A false statement is always taken to be not true, it is exactly what is not a true statement. Truth is what is the case, and false is what is not the case. They are the essential opposites. To think otherwise is absurd. 2+2=4 is true and 2+2=5 is false. That is what true and false mean.
There is no such thing as unintentional lying, because lying is entirely about intentionally telling a falsehood, which has a pronounced stigma of bad intent about it. If you tell a falsehood but believe it, that's not lying. You're still wrong, but you're not untrustworthy because of that. If you say that someone is lying, it is an accusation of a bad thing, not just a statement of how accurate someone is.
But maybe you're wondering if you say a belief for something that happens not to be the case but that you fully hold to be true, is that then true? No, that is not true. That is not what true means. A belief that is not true is a thing that is not true. A thing that is not true is not the case. Just because you believe it does not make it true, however strong your belief. (That's part of a good reason why we have distinct terms for such sentences, a belief is something you think is the case whether it is true or not, but a truth, whatever you may think, is still true.
That's just what words mean. To use them otherwise is perverse. If you understand words the way you want them to mean, then that may be easier for your own thinking, but you'll have difficulty in having other people understand you.
'True' is for things that are the case, and 'belief' is something you think is true (and may actually be true or not), and a 'lie' is an intentional non-truth, and these are three distinct things.
Of course, one can always read things metaphorically, and then maybe say that even false statements may have a grain of truth to them, like the quote from Picasso:
We all know that Art is not truth. Art is a lie that makes us realize truth at least the truth that is given us to understand. The artist must know the manner whereby to convince others of the truthfulness of his lies.
This is of course poetry, intentional mind bending, where we're supposed to see in an artificial, hand-made, inaccurate recreation of truth that lies behind the superficialities. A story that is made up but reveals truths.
And then there're are the universal truths which turn out to be myths that are malleable from one culture to another, form one point of view to another. Lots of historical truths are like this.
Also, using words to describe other words is circular. At some point thought has to come into it. And that is problematic because two people can point to the same thing and use the same words for the same thing but think slightly different concepts. It's even more tenuous for things you can't point at like abstract things like 'hope', 'identity', or even something which we all agree on like 'truth'.
This leads to change. Words certainly change how they sound over time, but the same sounds can change meaning over time. For example the word 'awful' used to mean more literally 'full of (or inspiring) awe', but now it just means 'terribly bad'. Meanings change for many reasons such as misuse, new contexts, weakening by overuse.
For example, 'literally' means 'by the letter' or 'verbatim' or 'based on the primary meaning'. But it is often used in a non-standard way for 'very much so' or 'virtually', mostly because the standard way of using it comes in contexts where 'virtually' would fit also.
I think what you are really asking is if there has been any change over the years in the use of the word 'truth', if there has been any change or split in the usage where there is still a standard but also a new non-standard use. I think you are suggesting that people (Americans) are starting to use 'truth' for things that are strongly held beliefs (I am editing what you said because it is a true oxymoron to use 'unintentional lying' because lies are by definition intentional).
There has been lots of talk in the news lately by people who seem to say things that are just not the case but they claim truth. For the record, it is not just lately, and it's not just Americans; politicians and used car salesman have been saying forever 'Believe me, I'm telling you the truth when I say ... ' (something that eventually turns out to be false). Saying 'X is true' can be judged for truth just as much as 'X'.
Let's just say that it is plausible that there has been some semantic drift, but because the meaning of 'truth' is so clear, this drift is not really happening. It is more likely that these handful of people are not sincerely using the words differently, but are intentionally labeling things wrong. That is, they are making a mistake in using 'truth' or just lying.
Side Note 1: Definitions from dictionaries are interesting. There was no god-like person who created definitions for words (except maybe in science and math and that's out of convenience). A dictionary attempts to describe in as few words as possible (that's important) what a word means using other words. Some dictionaries employ scores of linguists who are experts in capturing this meaning well. But the point is that what they try to capture is how other people, on the whole, tend to use a word. The dictionary writing linguists are not trying to tell you that you are supposed to use a word in one way and not another. A good dictionary will say that one meaning is the socially approved or formal or educated use (what we usually call 'correct'). But the dictionary is not an enforcer or It is an authority in the sense that the writers tend to know a lot about how to explain the many nuances of the word accurately. But it's not an authority like police who use punishments to enforce laws (still a cultural source), and it is certainly not an authority on what must be the case, like physical or chemical laws.
Side note 2: There are all sorts of competing philosophical definitions of 'truth' which mostly overlap but have some nuanced differences. None of these philosophical definitions include 'falsehood'.