Your spell slots recover based on the class feature that allows your spellcasting.
The language in the multiclassing section only mentions how you cast spells, not how you recover them. Note that recovery isn't mentioned at all in that paragraph. The spell slots are interchangeable when you're casting, but they recover based on their own mechanics. Unlike spell slots from multiple classes with Spellcasting, they don't merge into one pool; they remain separate except when you're actually casting out of them.
As a Warlock/Sorcerer, you have both the Pact Magic feature and the Spellcasting feature, which recover spell slots separately. Pact Magic slots recover on any rest, and Spellcasting slots recover on a long rest.
Short rests require an hour in which
a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
(PHB, page 186.)
Eldritch Master takes one minute, and doesn't specify what you can and can't be doing in that time. It's unclear (i.e. open to DM ruling) if you have to spend that minute doing nothing except entreating your patron, but even if that's true, sometimes, you don't have an hour. If you're running away from something, you might just be able to get a minute lead on it to quickly get your spells back. An hour lead is a lot less likely.
A minute is a very short time out of combat, and even in combat, it's 10 rounds. In a protracted fight you could probably pull this off. In a really protracted fight, you might need to. For example, if you're fighting the Tarrasque, your Eldritch Blast will be entirely useless. Once you're out of spell slots, you're going to be standing around doing nothing. (Slight exaggeration.) Your allies might be able to keep you alive for 10 rounds so that you can get some spells which have a chance of achieving something.
It shouldn't need pointing out, but an hour is 600 rounds, which is just not going to be possible in any combat. (Unless you're fighting a Dire Half-Dragon Fiendish Snail or something, but that's just getting silly.)
Of course, all of this is somewhat pointless if your DM is ok with you doing other things during the minute it takes to use this. For example, even if you're only allowed to use your movement, running around for a minute screaming at your patron for help seems completely achievable. If your DM rules that you can fight as normal during that minute, then this feature might actually be too powerful. Get into a fight with a tough enemy? Start praying to your patron. That gives you 10 rounds to use all your resources, at the end of which you just get them back immediately. Personally, I wouldn't be quite that lenient, but that's up to your DM.
Best Answer
The first link you provided is a homebrew (non-official) class named artificer.
The official artificer (published in Tasha's Cauldron of Everything and Eberron: Rising from the Last War) regains their spell slots after a long rest: