Tidal Warrior
Spawning Pool

Ok, so this question is regarding rule 305.7

305.7. If an effect sets a land’s subtype to one or more of the basic land types, the land no longer has its old land type. It loses all abilities generated from its rules text, its old land types, and any copy effects affecting that land, and it gains the appropriate mana ability for each new basic land type. Note that this doesn’t remove any abilities that were granted to the land by other effects. Setting a land’s subtype doesn’t add or remove any card types (such as creature) or supertypes (such as basic, legendary, and snow) the land may have. If a land gains one or more land types in addition to its own, it keeps its land types and rules text, and it gains the new land types and mana abilities.

Scenario is this:
Spawning Pool is activated to a creature. Tidal Warrior then targets it with its ability and that resolves.

Question:
Does the land look like this?

Spawning Pool
Creature Land - Skeleton Island
T: Add U to your mana pool
B, T: Regenerate ~this.
(is black)

I.e. it looses its ability to animate, but retains its granted ability of regeneration since that is not rules text of the card. Have I understood this interaction correctly?

Yes, you have. Your writeup of the Pool's appearance is slightly inaccurate, since you left out the power and toughness (1/1) and accidentally inserted a tap symbol that shouldn't be there into the cost of the regeneration ability, but you have the right idea.

Looking at how the abilities are applied in the layer system, it becomes clear why the regeneration ability remains: Tidal Warrior's ability removes the Pool's abilities as part of turning it into an Island in Layer 4, but that can't possibly remove the regeneration ability because at that point the Pool doesn't have the regeneration ability yet–the Pool will only gain that ability later, in Layer 6.