Turn Count Inconsistency when using Attack in Exploding Kittens
I have a very interesting answer to this question that might or might not be correct...
For all of this to make sense, I believe the first step should be to differentiate between a normal turn and an attack turn.
If you do that, all different scenarios and rules start slowly making more sense and matching up:
1) Official Rules:

ATTACK 2X 4 CARDS
Do not draw any cards. Instead, immediately force the next player to take 2 turns in a row. Play then continues from that player. The victim of this card takes a turn as normal (pass-or-play then draw). Then, when their first turn is over, it's their turn again.
If the victim of an Attack Card plays an Attack Card on any of their turns, the new target must take any remaining &lt;attack&gt; turns plus the number of attacks on the Attack Card just played (e.g. 4 turns, then 6, and so on).

See how I edited in a word there? Would it remove the contradiction from your examples?
1.1)

P1 plays Attack, so P2 has 2 turns. (0 normal and 2 attack)
P2 plays Attack, so P3 has 4 turns. (0 normal and 4 attack)
P3 plays Attack, so P4 has 6 turns. (0 normal and 6 attack)
P4 plays Attack, so P5 has 8 turns. (0 normal and 8 attack)
P5 plays Five Different Cards (to get Attack) and draws Shuffle, with 7 turns left. (0 normal and 7 attack)
P5 plays Attack, so P6 has 9 turns. (0 normal and 9 attack)

So far so good!
1.2 &amp; 1.3)

The inconsistency is that P1 already has 1 turn when playing Attack, which would imply adding 1 turn per Attack card

Now this inconsistency is gone! Normal turns don't get carried over, only attack turns!
2) Mobile Game
Another source for rules that gave me this idea is the mobile Exploding Kittens game.
The game has interesting type of cards called &quot;Slap&quot; (as a replacement for attack cards, functionality seems to be identical):

There are double slaps, triple slaps as well as single slaps and their behavior does seem to fit in with my answer:
Imagine you start the game by single slapping someone:

P1 (1 normal turn, 0 attack) -&gt; Slap
P2 (0 normal turns, 1 attack)

This is the behavior that you would expect according to my explanation but what if P2 slaps back?

P1 (1 normal turn, 0 attack) -&gt; Slap
P2 (0 normal turns, 1 attack) -&gt; Slap
P3 (0 normal turns, 2 attack)!

That's precisely what happens in the game! It seems like for the rules to make sense you really need to differentiate between normal turns that happen when someone finishes their turn, and the forced &quot;attack&quot; turns that happen once someone gets attacked/slapped.
2.1) Official Rules (again)
The game behavior and my explanation both match up with the &quot;Double Slap&quot; description as well:

Double Slap
3 Cards with  4 Cards without
End your turn without drawing a card and force any other player to take 2 turns in a row.
This is not the same as drawing 2 cards in a row! The target of this card should take their first turn (play cards and/or draw). When their turn is over, it’s their turn again.
If the targeted player plays a Slap Card in return, they pick a player to take ALL remaining turns (4 turns, then 6, and so on). Play always continues from the targeted player.
Triple Slap works the same way as Double Slap, but adds 3 turns instead of 2.

This description matches my explanation as well!

Whether this is the correct interpretation of the rules or not is up for debate, but this was the one and only way I have managed avoid any logical inconsistencies while playing the game both in real life and virtually and reading different rules (old rules are even more different!).