Is per-record timeline consistency is somehow equivalent to causal consistency?
You may know that per-record timeline consistency is used in PNUTS: Yahoo!’s Hosted Data Serving Platform:


  We provide per-record timeline consistency: all replicas of a given record apply all updates to the record in the same order.


Answer your questions:


  Since all modifications to a specific primary key are causally related (a client reads it, then modifies or delete it), ...


They are actually totally (or sequentially) ordered, stronger than "causally related". 


  ... then per-record timeline consistency is somehow equivalent to causal consistency.


Given the reason above, per-record timeline consistency is more like sequential consistency (wiki). In fact, per-record time consistency is also called per-key sequential consistency, which is stronger than per-key causal consistency. Because sequential consistency is not local, per-key sequential consistency is not equivalent to (i.e., weaker than) sequential consistency defined over multiple keys.