Your third interpretation is the way surprise works by the rules.
The rules on surprise state (emphasis mine):
The DM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the DM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn't notice a threat is surprised at the start of the encounter.
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't.
The surprise rules refer to the hiding rules, detailed in a separate sidebar:
The DM decides when circumstances are appropriate for hiding. When you try to hide, make a Dexterity (Stealth) check. Until you are discovered or you stop hiding, that check's total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence.
You can't hide from a creature that can see you clearly, and you give away your position if you make noise, such as shouting a warning or knocking over a vase. An invisible creature can always try to hide. Signs of its passage might still be noticed, and it does have to stay quiet.
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen.
Passive Perception. When you hide, there's a chance someone will notice you even if they aren't searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature's passive Wisdom (Perception) score, which equals 10 + the creature's Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.
The sidebar also references the rules on vision and light. Lightly obscured areas (e.g. areas of dim light) impose disadvantage on Wisdom (Perception) checks relying on sight when trying to see something in the area; heavily obscured areas (e.g. areas of darkness) cause Wisdom (Perception) checks relying on sight to fail automatically when trying to see something in the area.
(As noted, disadvantage on an active check is treated as a -5 modifier to the corresponding passive score.)
Special senses may impact how these areas are perceived - for instance, darkvision causes areas of dim light within its radius to be treated as bright light, and areas of nonmagical darkness within the radius to be treated as dim light.
Together, these two sets of rules explain how the surprise mechanics work by default. In summary, assuming the PCs are trying to surprise the enemies:
- PCs that don't attempt to hide at all will likely be noticed by
enemies close enough to see or hear them, unless the DM rules that
the enemies are sufficiently distracted or otherwise occupied to
notice any non-hiding party members.
- PCs attempting to hide must make a Dexterity (Stealth) check. This is
contested by the Wisdom (Perception) check of any enemy that is
actively searching for them (e.g. if they're on guard/patrol, or
otherwise have a reason to look around despite not knowing the PCs
are there), or the passive Wisdom (Perception) score of any enemy not
actively searching for them.
- If a PC's Dexterity (Stealth) check total is lower than an enemy's
Wisdom (Perception) check total (if the enemy is actively searching)
or passive Wisdom (Perception) score (if the enemy is not actively
searching), that PC is noticed by that enemy.
(This comparison is theoretically done for each hiding PC with each
enemy that could perceive them... That said, if the hiding PC with
the lowest Dexterity (Stealth) check total is within sight/hearing
range of all enemies, then there's no point comparing the other PCs'
totals; either the enemies notice that PC and aren't surprised, or
they fail to notice that PC and are surprised.)
- If an enemy notices any threat at all (whether the PCs or a third
party) when combat begins, that enemy is not surprised.
Note that initiative is rolled at the beginning of a combat encounter before any combat actions are taken; the effects of surprise take place during the first round of combat. If an enemy notices a PC and perceives them as a threat, this almost always leads directly to combat. Alternately, if none of the PCs have been noticed but a PC intends to harm an enemy or target them with a spell, initiative must be rolled, and such attacks/spells/etc. would need be done on the PC's turn as part of the first round of combat.
In essence, "surprise" means the enemy is not ready for a fight when combat begins. Thus, an enemy can not be surprised mid-fight; if an enemy sees one PC and is fighting them, but fails to notice a second PC, the enemy does not become surprised when the second PC reveals their presence.
There is still a benefit to being hidden from your target even if they're not surprised, however; the rules on unseen attackers and targets state:
When a creature can't see you, you have advantage on attack rolls against it. If you are hidden — both unseen and unheard — when you make an attack, you give away your location when the attack hits or misses.
Creatures that perceive any threat will continue responding to that threat as normal, whether or not there are other potential threats they don't detect.
Regarding group checks and other variants
I haven't been able to find any other official variant rules or suggestions around how to resolve surprise, either in the Player's Handbook, the Dungeon Master's Guide, or Xanathar's Guide to Everything.
Some DMs may use group checks and/or other house-rules in determining surprise. However, the core books never seem to suggest using group checks for this purpose, effectively making the specific use of group checks for surprise (a mechanic whose rules are already clearly defined) a house-rule as well. The rules on group checks state (emphasis mine):
When a number of individuals are trying to accomplish something as a group, the DM might ask for a group ability check. In such a situation, the characters who are skilled at a particular task help cover those who aren't.
To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. Otherwise, the group fails.
Group checks don't come up very often, and they're most useful when all the characters succeed or fail as a group. [...]
Note that it mentions that group checks are more useful when success/failure is determined on a group level rather than for individual characters, whereas the surprise rules specifically note that a member of a group can be surprised even when the other members aren't. In addition, a number of class features or magic items specifically prevent the character that has them from being surprised; most of these don't extend this effect to the party as a whole. As such, group checks are not a great fit for determining surprise.
Nevertheless, some DMs use them anyway, generally to cover for parties that want to be stealthy but have one or two members that are terrible at Stealth. For instance, Dan Dillon, an official game designer for D&D, explained his house-rules involving group checks for surprise in this series of tweets from June 2019:
When the party sneaks and the DM uses a group Dexterity (Stealth) check, you take the result of the check and compare it to all observers’ passive Perception scores. They notice or fail to notice the party as a whole.
(He elaborates on the details in subsequent responses which I've summarized below.)
Dillon reiterated the house-rule in this tweet just over a week later:
So when I do group checks for Dexterity (Stealth), I throw out the bottom half of the results, and use the lowest of the top half as the group’s check to see if they surprise the monsters.
In essence, all the PCs make a Dexterity (Stealth) check, but only the median result is compared to enemies' Wisdom (Perception) checks or passive scores. That median result determines whether the party as a whole is noticed by a particular enemy or surprises that enemy.
Note that this group checks house-rule involves exactly as many rolls as the official method. It theoretically reduces the number of comparisons that need to be made between PCs' Dexterity (Stealth) checks and enemies' Wisdom (Perception) checks/passive scores - but in most cases, the RAW method may as well only care about the lowest Dexterity (Stealth) check anyway, in which case there are exactly as many comparisons. This house-rule also adds an extra step of determining the median Dexterity (Stealth) check result before those comparisons are made.
As a result, this house-rule is not really easier or more efficient than the official rules on surprise; all it does is cover for some party members being bad at Stealth.