During the proof of work process, a miner receive a block to mine.
He doesn't "receive a block to mine" , he crafts a candidate block with a Coinbase transaction and with transactions aggregated from the mempool.
He mines it, whether the miner is rewarded or not, this block is added to his own chain.
This is false.
Miners do not have their own separate blockchain from the rest of the network.
Also, the miner is rewarded iff his block is valid according to the consensus rules; he finds a nonce when hashed with the Merkle root, timestamp, etc, gives a block header hash lower than the target hash; no other miner releases a completing block (valid block with the same block height).
When all the conditions are satisfied, then the Coinbase transaction he crafted (including the block reward) becomes valid -- and can only be spent after 100 blocks.
So at once, his has a chain which is different from the main chain.
He doesn't have a chain, just a block which will either be added to the main chain or not depending on if he follows consensus rules.
At this time could we say there are as many btc chains as there are miners ?
Any block that hasn't been added to the blockchain IS NOT part of the block chain; it's just a candidate block.
So they are many CANDIDATE BLOCKS but just one blockchain.
Next, is my understanding correct ?
No, not really.
The miner looses against miners, the block becomes orphan,
His block only gets orphaned if after he published his valid block, another miner publishes a block with the same block height but higher cumulative difficulty, if he doesn't find a block in the first place then he's not part of the blockchain and so his block can't get orphaned.
and the miner's chain reverts back to the main chain (somehow) and he waits for the next block to be mined.
Miners mine on the TIP of the main chain.
Miners craft candidate blocks on top of a block that's on the blockchain.
Until a miner finds a valid block and publishes it and other miners mine on top of it as the TIP of the next block, his candidate block is NOT on the blockchain.
For better understanding of the subject read Andreas' chapter on
mining in his book Mastering Bitcoin.