What distinguishes a valid 'pool' as a destination for rainwater versus a 'vessel' which would invalidate the rainwater?
If it is already defined as a vessel (from uncleanness point of view) before they fixed it to the ground. Anyway, if there was a hole great as a Bota bag-neck, or they did make afterward a hole (before filling with water), in the bottom, this is not a vessel because it can not be used as a container.
May the pool be entirely supported by the house floor?
Yes, if it is fixed, it is good enough.
must it be sunken into the ground?
No, the criterion is fixed or not fixed (sealed)
or may it merely touch ground by some means?
Touching is not enough. We need to fix it e.g. by lime or cement
Must the pool be made of one entire material, such as concrete or ceramic?
No, the unique criterion is to be fixed and not be a vessel before fixing.
or may it (to reduce weight) be a lighter substance which is merely lined with caulked ceramic tile?
No problem. The halacha tells anout wood
Gemara Baba Batra 65b:
Our Rabbis taught: 'If a man hollows out a pipe and then fixes it, water from it makes a mikweh unfit for use.
Mishna Para 5, 7:
A trough in a rock [in which water gathers from nearby spring], ... and it does not invalidate a mikvah [if water gathered in the rock trough and then flowed into a mikvah, that water is still considered undrawn, and thus valid for a mikvah, which must be filled with water which has never been drawn in order to be valid for ritual immersion and purification].
If a vessel was attached [to the ground] with plaster, we may fill [water for sanctification] with it, and we may sanctify [water] in it, and we may sprinkle from it, and it requires a 'tight-seal' [to protect its contents from impurity], and it invalidates a mikvah [if water flowed from it into a mikvah, it is considered to be drawn water and invalid].
If [it was punctured] from the side and stopped with a cloth, the water inside is valid, since they are encircled by vessel [note: this only works if the puncture is sufficiently elevated from the bottom]. [If] they made for it [around its rim] a crown of plaster [for the vessel attached to the ground] and the water [level] reached to there, it [the water] is invalid [for the ritual, since the crown is not considered a vessel].
[If] they made for it [around its rim] a crown of plaster [for the vessel attached to the ground] and the water [level] reached to there, it [the water] is invalid [for the ritual, since the crown is not considered a vessel]. If [the connection] is sufficiently strong such that lifting it [the crown] would cause the vessel to come with it, it is valid [since it is part of the vessel].
Halacha's book, the Shulchan Aruch.
YD 201, 6 and 7.