So i think this PDF is just a simple copy and paste thing from other Websites and not really your one work.

I was thinking about accepting Tether as a payment currency. But that would make it more complicated for me since in that case I would need to have wallets on all different blockchains. Also sender needs to know to which wallet to send the coins

Yep, correct. Since both blockchains are isolated on its own network, inter-transaction of USDT are not possible.

Another scenario is A send USDT on BEP20 chain to B, but B is expecting to receive it on the ERC20 chain. Due to the nature of the underlying technicalities, the transaction will be accepted but the transaction is only happened on the BEP20 chain, not on ERC20, thus B did not receive the token.

First of all, note that tether is not ethereum. Tether (or USDT) is a token that has been issued on many blockchains.
One of the blockchains in which you can make a USDT transaction is ethereum.

All USDT transactions that are made on ethereum blockchain can be seen on any ethereum explorer.
https://etherscan.io/token/0xdac17f958d2ee523a2206206994597c13d831ec7

I would recommend having a read of this page: https://learnmeabitcoin.com/beginners/outputs. It very clearly breaks down the concepts I described above, that you can only spend an entire output (or multiple outputs), if you don't want to spend the full value to the other party then you need to specify to send some back to yourself as change, and whatever is left over that you haven't specified a destination for gets used as the fee.

I would also suggest, though, that since you don't have a firm grasp yet on how outputs and transactions work, that you should use a client such as Electrum which will do all this for you in the background, rather than trying to build a transaction manually and ending up losing large amounts of coins.

Correct. Whatever amount of bitcoin you don't specify an output for will be consumed as a fee.

In the first transaction you linked, your input was 8000 sats and you only specified a single output of 2000 sats, so the remaining 6000 sats was used as a fee. If you did not want to spend all that on a fee, then you should have specified an additional output to a change address of say 5500 sats, leaving 500 sats unaccounted for which would have been taken as the fee.

Remember that if you try to send 2000 sats, bitcoin cannot "skim" 2000 sats from an available input. It must spend the entire input, and so you must specify a destination for all the coins in that input you do not wish to be spent as a fee.

I was thinking it could be because of the transaction weight before but by tracking the txid, you are very right,  adamcro definitely customized the fee.

Or maybe he is using a non-reputable testnet wallet, but I doubt that because the fee using testnet is very low.

Which wallets are you using? See if they allow to customize the fee.