Are there some foolproof methods of differentiating the originals from the interpolated texts in such cases? If yes, what are those?
There are some accepted methods explained further below but they are not perfect.
Are those methods mentioned in Hindu scriptures themselves?
For how the integrity of the Vedas is preserved, see this answer. I'm not aware of any such technique to preserve Purāṇas, Itihāsas and other Smṛtis.
If you read the report titled Interpolations In The Mahabharata by Prof. M. A. Mehendale of BORI you will get an idea, how scholars, in general, identity and remove spurious material from Purāṇas and Itihāsas.
Although the report is more focused on The Mahābhārata, which is an itihāsa, I'll quote below some lines from the report which are applicable to Purāṇas as well. Mehendale is saying that, no matter how hard you try, you cannot get the scripture to its original form. So even if see read the words "critical edition" in the title of a book, read it with a grain of salt!
The critical edition of the text now lays bare all spurious matter which could be easily set aside on the objective evidence of the manuscripts. The guiding principle that was adopted for this initial task of sifting was not to admit any passage, short or long, in the constituted text if that was not uniformly documented in all the versions of the Northern and the Southern recension.
...
The task of removing the spurious matter that still remains in the critically constituted text and bringing it as close as possible to the oldest version of the Mbh. has been left by them to future students of the text.
For, arriving at the truly "original" text of Vyasa is well nigh impossible.
Below, he lists the 3 different types of interpolations.
What I propose to do this evening is to give some examples of different kinds of interpolations in the Mahābhārata.
- Those revealed by manuscript evidence and set aside by the editors,
- those that are not revealed by manuscript evidence and hence form part of the constituted text but are quite obvious due to contradictions in consecutive stanzas, and
- those revealed either by contradictions that are not so obvious because they are not found in consecutive passages but are found in the text as a whole, or by the criterion of intrinsic probability.
Following all the above three methods, one would assume the critical edition minus all the interpolations will give us the perfect text everyone can read. But unfortunately there maybe some cases where certain text is purposely omitted or deleted from the manuscripts and in those cases it's difficult to reconstruct the true account of a story from bits and pieces of information from other parts of the manuscript.
Examples of Interpolations
From The Mahābhārata:
One such example is related to the svayaṁvara of Draupadi. It is
popularly believed that when Karṇa lifted the bow and was about to shoot the
arrow to hit the target. Draupadi loudly declared that she would not choose
a sūta for her husband: drstvā tam draupadi vākyam uccair jagāda nāham
varayāmi sūtam (1.1827, p.725). On hearing it, Karṇa threw down the bow and, downcast, retired from the contest.
This stanza does not occur at all in the entire Southern recension, and
among the versions of the Northern recension, it does not occur in the
Kashmiri, Maithili and Bengali versions. It is found only in four
Devanagari mss. (out of a total fourteen used for the critical edition), and one
(out of three) Nepali ms. It is therefore clearly a very late addition to the
text.
I discuss the above interpolation in my answer to Do scriptures indicate that Draupadi regretted not allowing Karna to participate in her svayaṁvara?
Here's one from Uttarakāṇḍa of Vālmīki Rāmāyaṇa:
