There is no such thing as "instant transactions" in bitcoin. You are talking about "unconfirmed transactions".
In bitcoin when you send a transaction, almost always it propagates to all nodes within seconds. These transactions reside in nodes memory pool (assuming they remain valid). During this time they are called "unconfirmed" and they can be double spent which is why it has never been safe to accept an unconfirmed transaction.
When the next new block is mined (which could take from a second to a couple of hours) it could contain that unconfirmed transaction if it was paying a high enough fee compared to others or if the mempool didn't contain enough higher paying transactions to fill the block. This is referred to as being "confirmed" and as you said it becomes increasingly harder to reverse these transactions.
For higher amounts, e.g. US$1,000, it makes sense to wait for 6 confirmations or more. Any confirmation can reduce the risk of reverse transactions exponentially.
There are a couple of things that are usually ignored by people when talking about number of confirmations.
1. The network state.
Under normal circumstances (eg. right now) there is nothing going on so the chances of a chain split and a reorg is minimal but during a fork or a network disruption like 2017 or anything that could increase this risk the number of confirmation one demands has to increase. In some cases you may even wait for more than 200.
2. The client type.
A full node will be aware of any chain split or generally speaking any form of disruption in the network. An SPV client on the other hand may not even figure it out for a long time and remain on the wrong chain.
There is also more "centralized" non-custodial wallets where they only rely on a single server (most of phone wallets) that have a much higher risk.
In cases like this the number of confirmation required for security is much higher.
3. The amount and the payer.
Finally the only factor people are aware of. For higher amounts you want more confirmation and vice versa. Also if you know who is paying you (eg. a friend sending you money he owed) you may require a different number of confirmation. For example a food truck doesn't care about confirmation because the risk of someone double spending a bitcoin payment for a hotdog is the same as the risk of someone giving them fake cash bills, it's the acceptable risk of running a business.
Speaking of amount, it is not arbitrary. It also relates to the cost of the attack as @ranochigo pointed out. Since in order to reverse a confirmed transaction you'll have to perform a 51% attack and the cost of that is extremely high, $1000 can not be considered a high amount.