the important thing you have to keep in mind is that a "private key" is universal. which means if you store the private key like this:
it will work forever and in any wallet application. it doesn't matter how you created it as long as it was a safe way, it will work and it won't change.

but when you are talking about words aka seeds or some call it recovery phrase, in some cases the wallets may implement their own specific method of getting the keys from the seeds. for example
this is an Electrum seed. it will not work on other wallets and in order to recover your coins you need to either use Electrum or modify the code in another application/tool in order to get the keys and be able to use them.

it is not a big deal since all of the good wallets like Electrum are open source but it is best to know it now rather than having to figure it out and dealing with it later. and also you need to check what the wallet you are using (the web wallet you mentioned) is using for its Seed generating method.