Sorry this wrong. Motherland does depend only on TC. It is essentially the ratio of your TC in the city versus the TC of one or more other civs.
Let's consider an enemy city that has been owned for a long time. It has over 500 CC (we can see this since it has 60% defence) Let's consider two situations, one where we just capture the city (case A) and another where we run several spy missions and raise our CC in the city to 500 (case B). The city already has our state religion.
In case A when the city comes out of revolt it it will have 0 CC and a Cpt of 1 due to the religion. There are no border pops. We start building a theatre and whip it in immediately, it will complete next turn. For TC; the city is putting 1 culture into its city tile and the surrounding zone.
In case B when the city comes out of revolt it it will have 500 CC and a Cpt of 1 due to the religion. There are 3 border pops. We start building a theatre and whip it in immediately, it will complete next turn.
For TC case B:
The city tile will get 1 Cpt plus 3 zone bonuses of 20 culture = 61 TC
The zone 1 tiles also get 61 TC
Zone 2 tiles get 41 TC
Zone 3 tiles get 21 TC
Zone 4 tiles get 1 TC
--------------
The Next Turn. The theatre completes and we run 2 artists and start building a library.
Case A (turn 2)
Cpt = 1 (religion) + 3 (theatre) + 8 (2 artists) = 12
CC = 1 (last turn) + Cpt = 13 (so border will pop next turn)
City Tile TC = 1 (last turn) + Cpt = 13
Zone 1 TC = 1 (last turn) + Cpt = 13
Zone 2 TC = 0
Case B (turn 2)
Cpt = 1 (religion) + 3 (theatre) + 8 (2 artists) = 12
CC = 501 (last turn) + Cpt = 513
City Tile TC = 61 (last turn) + Cpt + 3 zone bonuses = 61 + 72 = 133 TC
Zone 1 TC = same as city tile
Zone 2 TC = 41 (last turn) + Cpt + 2 zone bonuses = 41 + 52 = 93 TC
Zone 3 TC = 21 (last turn) + Cpt + 1 zone bonus = 21 + 32 = 53 TC
Zone 4 TC = 1 (last turn) + Cpt = 1 + 12 = 13 TC
--------------
The next turn. The library is still building in bothe cases. The two artists are still being used. Case A has its first border pop.
Case A (turn 3)
Cpt = 1 (religion) + 3 (theatre) + 8 (2 artists) = 12
CC = 13 (last turn) + Cpt = 25 (Border popped this turn)
City Tile TC = 13 (total last turn) + Cpt + 1 zone bonus = 13+12+20 = 45 TC
Zone 1 TC = same as city tile
Zone 2 TC = 0 (last turn) + Cpt = 12 TC
Case B (turn 3)
Cpt = 1 (religion) + 3 (theatre) + 8 (2 artists) = 12
CC = 513 (last turn) + Cpt = 513 +12 = 525
City Tile TC = 133 (last turn) + Cpt + 3 zone bonuses = 133 + 72 = 205 TC
Zone 1 TC = same as city tile
Zone 2 TC = 93 (last turn) + Cpt + 2 zone bonuses = 93 + 52 = 145 TC
Zone 3 TC = 53 (last turn) + Cpt + 1 zone bonus = 53 + 32 = 85 TC
Zone 4 TC = 13 (last turn) + Cpt = 13 + 12 = 25 TC
--------------
We can see that case B gets a much stronger start both in removing the Motherland in its city tile and in spreading TC over a wider area. Tile culture in the city tiles after 3 turns is 45 (case A) versus 205 (case B) and it is going to get the extra zone bonusses for a long time. Any neighbouring cities in Zone 3 or 4 is going to get an immediate benefit in case B but has to wait for the CC to increase in case A.
Lots of mental arithmetic there
I'll leave it for someone else to check it for mistakes.