The core distinction lies in the operational framework and the level of custody involved. The Fiat channels proposal operates on a purely custodial basis through the use of hosted channels. This method allows for a proof of misbehavior, enabling users to demonstrate instances where the host may have acted inappropriately. However, this form of proof is considered weaker compared to the unilateral enforcement mechanisms inherent to the LN itself.

On the other hand, Tony's construction maintains its operation within the standard parameters of a normal LN node, using channels denominated in BTC. This setup preserves the user's control over their assets, with the primary risk being the peer's failure to cooperate in updating the exchange rate or adjusting holdings accordingly. In such scenarios, the user is left with BTC that may not reflect the current exchange rate, but they retain possession of their assets, unlike with hosted channels where the outcome could result in total loss.

Another critical difference is the flexibility offered by Tony's construction in integrating both BTC and USD channels within the same LN node. This integration not only facilitates the direct mixing of currencies but also enables the provision of swap services directly from the node. This contrasts sharply with the hosted channels' approach, where clients do not operate as normal LN nodes, thereby limiting their functionality and utility.

The comparison raises important considerations regarding how solutions within the LN ecosystem are categorized and discussed. Emphasizing the distinct features and trade-offs of each construction is crucial for a fair and accurate representation of the landscape. This discussion highlights the importance of recognizing the diversity of solutions available and the need for precise language when comparing them, to avoid overlooking significant differences that could influence the development and adoption of LN technologies.