No, a 3rd level sorcerer can't start with 4 2nd level spells
Let's begin with the "Starting at higher level" section (DMG p. 38):
Creating a higher-level character uses the same character creation steps outlined in the Player's Handbook.
Such a character has more hit points, class features, and spells, and probably starts with better equipment.
However, the PHB describes only how to create a level 1 character, and what happens each time it levels up.
Each class with arcane spellcasting (that has to learn spells, opposed to the divine spellcasting classes, which simply know all spells) has a section called "Spells Known of 1st Level and Higher" that describes how spells are learned; this section has the same structure for all arcane spellcasting classes - the only differences being that some subclasses have a limitation on the spell schools, and the example varies based on full/half/third progression.
For example, as a sorcerer:
The Spells Known column of the Sorcerer table shows when you learn more sorcerer spells of your choice.
Each of these spells must be of a level for which you have spell slots.
For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.
Additionally, in the "Spells Known and Prepared" of the multiclassing spellcasting section (PHB p.164), the following example is given, which implies that leveling up is done progressively:
If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class.
As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells.
So it's reasonably safe to assume that characters that start at higher level are created as if they started at level 1 and then incrementally leveled up.
What does this mean for your sorcerer?
- at 1st level, you know 2 1st level spells
- at 2nd level, you learn one additional 1st level spell, and can, if you want, change one of the previous known spells into another (still 1st level) spell
- at 3rd level, you learn one spell that can be 1st or 2nd level, and can, if you want, change one of your previous known spells into another (1st or 2nd level) spell
So at 3rd level, you can end up with at most 2 2nd level spells.