My friend and I are using LTN properly for the first time. 50 hours into space exploration we have 150-ish train stations. We understand how the thresholds work and that a station is only registered as a provider when the threshold is surpassed. This means that the request stations start spamming that there is no station supplying when the demand is not covered immediately even though there is a station that will be able to supply at some point. Sure we can suppress the warning globally or at every request station but that doesn't seems very intuitive as I would still like to know when there really is no station in the network that supplies the corresponding item.
I think requester should differentiate between there is no station and there is not enough supply.
i might also just use the mod incorrectly, but i am not aware how i should use it then.

That's just the way it works. If you connect up a demanding station without having a valid supply station for that demand, you will get the warning. A valid supply station means one that a train can be sent to, so it requires that the station exceeds the supply threshold.

My advice is to just disable warnings at the requesting station until you have established a reliable surplus of supply.

if you don't want to see the warning messages and if you use the LTN combinator (https://mods.factorio.com/mod/LTN_Combinator_Modernized), there is a switch to turn off the warning messages (bottom right).

After your initial supply and demand loop is well-established, with a solid surplus of supply, then you can turn that warning back on. After that it serves its intended function of warning you that something changed and you no longer have enough supply to meet the demand.

On the other hand, if you have a station that you always want to have more demand than there is supply (such as a station which recycles unwanted byproducts), then just leave the warning turned off.