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.
For your specific character, you are correct.
However you left off the invocations (which can be quite invaluable and often act as a spell without needing a spell slot). I'll break this down specific to your character:
Cantrips: 2 base +3 from Pact of the Tome = 5 cantrips (+1 from Infernal Legacy covered below)
Spells: 4 known + Hellish Rebuke - these spells must be selected from the Warlock spell list. The Great Old one patron allows you to select from an expanded spell list which includes the spells listed under Great Old One on PHB pg. 110. This means you can select from these spells when you learn a new spell, not that you know them in addition to the ones you do select.
Invocations: 2 invocations, these do not count against spells known. There is no requirement to know the spell in advance, otherwise this would be addressed under prerequisites in the invocation. In fact, the Invocations have a different name, and you can infer that the name of the Invocation is the Warlock version of said spell, with the spell name under said Invocation being used as reference to it's capability. This is demonstrated through Invocations such as Armor of Shadows which is effectively Mage Armor without the spell slot or material component requirement. Another example is Book of Ancient Secrets, which besides letting you pick any 2 rituals from any class (wow!), it allows you to cast a lot of your Warlock spells as rituals so you don't have to expend a spell slot (which are invaluable as a Warlock).
Infernal Legacy: All spells known from this are separate and distinct from your spell slots and invocations. These do not use spell slots to cast but function on a specific recharge as indicated by the description under Infernal Legacy. This is separate and distinct because it is not a Warlock feature, but a Race feature and does not require spellcasting or pact magic in order to utilize (it's in effect the same thing as a Dragonborn breath weapon).
Best Answer
Warlocks are a bit different from other classes, so it's fair that it can seem confusing at first!
Spells
Cantrips Known
These are the number of cantrips your Warlock knows. Cantrips do not cost Spell Slots and can be used almost every turn.
Spells known:
These are the total number of spells known for the Warlock at the proposed level. When a warlock wants to spend a Spell Slot to cast a spell, they have to choose from the spells they have Known.
Keep in mind, this is separate from Cantrips Known.
Spell Slots
This is how many non-cantrip spells a Warlock can cast per Short Rest. Once you take a Short Rest, you get all of these back.
Slot Level
This is how big your slots are. All of your Spell Slots for a Warlock are cast at this level. In DnD 5th edition, many spells can be "upcasted" (casting them at a higher level than is required) for additional benefits. A spell like Hex (Level 1 spell) has additional benefits when you cast it with a spell slot that's larger than level 1, which will happen often as a Warlock.
Invocations
Invocations are different from spells. These are special permanent benefits you choose for the Warlock, in its own section of the Warlock class. Example invocations include being able to knock enemies back with the Eldritch Blast cantrip, being able to see in magical and nonmagical darkness, and being able to turn invisible.
An example scenario could be a level 5 Warlock. They would have:
Another way you can look at it is that this Warlock has two level 3 spell slots, and regenerates those two spell slots each short or long rest.
Why Warlock?
Warlocks lack versatility in that they do not have many options for what to cast and how often to cast them compared to other casters (the Wizard can cast a total of 9 spells at level 5, and can cast all of them in a minute) but the spells that Warlocks DO cast are at very high levels. Note that a level 5 Wizard can only cast level 3 spells twice per long rest, where the Warlock can cast that many per short rest.
Assuming there are two short rests per long rest, that means that the Warlock can cast their highest powered spells 3x more than the Wizard.
A Wizard may excel in versatility, but nothing compares to the Warlock in raw, consistent firepower.