How can I make a yogurt starter culture from scratch?

The question may be conflating two different things. First, let's be clear about what yogurt is: the word traditionally refers to a milk product produced by fermentation with some lactic acid bacteria strains. (The exact strains of bacteria may vary depending on the culture and method, though the word "yogurt" tends to be restricted to thermophilic bacterial strains which ferment best when milk is above room temperature.) The yogurt becomes sour through the production of lactic acid, which is created as a waste product from the bacteria as they grow in the milk ("fermentation").

The second aspect of yogurt beyond the sour flavor is the texture. When milk's pH is reduced to a certain point through introduction of acid, the proteins in milk unfold (denature) and begin to clump together. The milk thus begins to "curdle" and take on a thicker texture which is generally associated with yogurt.

Any source of lactic acid bacteria can potentially be used to create a starter to generate yogurt. In India, chili peppers and particularly their stems are frequently used, as they tend to harbor several bacterial strains that can ferment milk. The resulting product is a type of yogurt, known as dahi or sometimes simply called curd in English.

Dahi can therefore be considered a type of yogurt, produced through fermentation of lactic acid bacteria at warm temperatures.

However, the OP also mentions the introduction of lemon in milk and links to a previous question which states that this can produce "cheese" rather than "yogurt." The reason for the difference is that the method of production is very different with lemon juice. Usually, when people just add lemon juice to milk, they are trying to artificially sour it quickly. The lemon juice will lower the pH, producing a sour flavor. It will also cause the proteins to denature and the milk to thicken. This process happens in the matter of a few minutes, rather than hours.

The resulting product is superficially similar to cultured yogurt or buttermilk in sourness and texture, but no fermentation has taken place. No lactic acid bacteria were involved. The resulting product here is simply what I'd refer to as "sour milk" or "soured milk" or "acidified milk." Because there is no fermentation, and the acid flavor comes from lemon juice (with citric acid) rather than lactic acid, it's not what most people would refer to as "yogurt." And, as the linked answer notes, this process usually won't produce a very thick yogurt-like product unless you use a milk with high fat content like buffalo milk.

In any case, I personally would not call it "cheese" unless you actually strained the resulting soured milk. But it's not "yogurt" either, according to traditional definitions: it's merely acidified milk.

In sum, both chili peppers and lemon juice can be used to sour and thicken milk, but the process is a bit different and produces somewhat different products.