The previous Mishnah dealt with a case of doubt that does not relate to an event that took place in a private domain or in a public domain, in a way that would let us establish a clear rule whether to be stringent or lenient. Rather, the doubt itself contains within it the question of whether the tumah was in a private domain or in a public domain. That is how we explained the previous Mishnah.
It is worth noting that some explain the previous Mishnah as referring to a slightly different case, though the point is identical - the question of whether we are dealing with a private domain or a public domain is included within the doubt itself: the source of tumah is lying in the public domain and the person is standing in the private domain, and a doubt arises whether he entered the public domain and touched the tumah, or the reverse. Here too the doubt includes both domains together, and the question is how to define it - a doubt that is more in the public domain or more in the private domain.
An example of this: a person standing on the ground with his head among the branches of a tree. The upper part of the tree is considered a private domain, at least with regard to the laws of tumah, and so the person's body is in the public domain while his upper part is already located in a private domain - and the very same argument stated in the previous Mishnah applies here as well.
The beginning of our Mishnah - a doubt that is entirely in a private domain:
"Ha'oleh l'rosh ha'ilan" - One who climbs to the top of a tree: a tree standing in the public domain with a source of tumah in it, and the person climbed all the way to the top. Since he is not standing in the public domain, but is entirely in the upper part of the tree, which is considered a private domain, if a doubt arises whether he touched the tumah or not - his doubt is ruled to be tamei, following the law of a doubtful tumah in a private domain.
"Hamachnis yado lachor shebakosel" - One who inserts his hand into a hole in a wall: a person is standing in the public domain, where many people are found, and he inserts his hand into a hole in the wall that contains a source of tumah. Since it is not the way of passersby in that place to insert their hands into this hole, the hole has an independent status of its own and is considered a private domain. Therefore, if a doubt arises whether he touched the tumah or not - he is tamei out of doubt.
We learn from this that the definition of a private domain with regard to the laws of tumah is a broad one, and even the top of a tree or a small cavity such as a hole in a wall is considered a private domain for this purpose.
Cases in which both domains are involved in the doubt:
A doubt whether he entered the shop: a doubt arose whether the person entered into the shop and became tamei. This doubt is ruled to be tahor, and this is according to the opinion of the Sage in the previous Mishnah, that in cases where the scenario of the doubt involves both the public domain and the private domain, we treat the doubt as though it occurred in the public domain, and therefore it is ruled leniently.
An object lying at the edge of the shop: an object that transmits its tumah only through contact, and it is lying at the edge of the shop, and a doubt arose whether a person touched it or not. Included in this doubt is also the question of whether he entered the private domain or not, and therefore such a doubt too is ruled to be tahor.
By contrast, when there are two shops standing, one tamei and one tahor, and a doubt arose whether he entered the tamei shop or the tahor shop - here he certainly entered a private domain, and the entire question is which private domain he entered. Therefore even the Sage of the previous Mishnah agrees that the doubt is decided stringently, as tamei, since it certainly occurred in a private domain.
The roots of the law:
As we have learned, the reason we are lenient with a doubt that arose in the public domain is because of the rule of chazakah: the status of a thing remains as it was until we know as a certain fact that it has changed. Therefore, when there is a doubt whether a particular item became tamei or not, we have no certain fact before us, and we have no power to view the status of the object as having ceased to be tahor.
On the other hand, the law of a doubtful tumah in a private domain is derived from the sotah - a woman who was seen alone in a private place with a man to whom she is not married. From there we learn that if the doubt occurred in a private place, we do not apply the concept of chazakah at all. Our Mishnah teaches that since we do not apply the concept of chazakah at all, it turns out that even when many different doubts are involved - we still rule stringently.