Submarine swaps don't provide any additional privacy than what's already available in the Lightning Network and Bitcoin blockchain. So you can increase the privacy of your Lightning network by routing through multiple channels such as onion routing.

From the above two options, the 1 seems to be more privacy due to less exposure.

A few points I haven't understood:

Do you mean that the process of Submarine Swaps does increase bitcoin privacy, similar to those 3 methods mentioned in the other thread ?

When I send BTC from exchange to Muun wallet, Muun wallet will open a channel. But then does this channel close itself automatedly after the transaction is done ? 

Additionally, there're also 2 ways to send BTC from exchange to Muun wallet, are there any differences ?

1. Send BTC directly to Muun wallet

2. Send BTC via exchange's Lightning network (Many exchanges now support it) to Muun wallet

Right, but the Swap process enables on-chain BTC to become off-chain BTC, wouldn't this action get rid of traces ? Furthermore, I swap the off-chain coins back to on-chain coins again before spending it. Sounds that the BTC got mixed enough

Regarding the 'routing' you mentioned, I use Muun wallet, how do I route through multiple channels like onion routing ?

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Right, so in order to have better privacy in Lightning Network, I would need to close the channel each time I finish a transaction ?  

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Yeah, I have Electrum for normal BTC transaction, but in terms of Lightning, Electrum seems more complicated to set up. I'm not too sure but do you need to send in BTC already in Lightning network, which doesn't increase the privacy Lightning already has. It's the Submarine Swaps that potentially make coins harder to track

Mod edit: Consecutive posts mergesd

I haven't used Muun in the past and therefore can't contribute to specific properties of this wallet.

I will try to add my opinion in a hopefully more generalized Lightning context.

When you open a Lightning channel there's a funding on-chain transaction that has ties to your wallet's past transactions and the privacy largely depends on how you used and combined past UTXOs in your wallet. This funding transaction defines the amount of coins that can move back and forth in the Lightning channel. At first your channel has almost 100% outgoing capacity but no incoming capacity.

A Submarine swap-out (LN-->on-chain BTC) to my knowledge, please correct if I'm wrong, now breaks the connection from the funding transaction to the receiving bitcoin address. All that's visible is an on-chain transaction from the Submarine swap provider to your receiving address. The received amount of course can't be greater than the channel's sending capacity.

After the Submarine swap-out your channel has some incoming capacity and less outgoing capacity. This can come handy to break the connection of another transaction to your wallet.

Via Lightning or Submarine swap-in (on-chain BTC-->LN) you could now receive coins up to the amount of incoming capacity of your channel whereever those originate from. When you perform another Submarine swap-out to another address of your wallet there would be no visible on-chain transaction connection of the coin flow (the swap-out would be again some possible merged multi-client on-chain transaction into a new address of your wallet).

With Taproot the channel's funding on-chain transaction would not appear as a Lightning channel funding transaction, which is a welcome privacy feature to hide the opening of channels on-chain.

When you close the Lightning channel there will be an on-chain transaction that has a connection to your funding transaction and it will be visible how the channel partners settled their end result of intermediate coin flow of the channel.

Submarine swap-out and swap-in don't come for free as the Submarine swap provider takes it's fees for the service.