This is a case where botany and English language overlap. To understand the sense in which the word 'pepper' is used in example sentence and to understand the relationship between 'pepper' and 'capsicum' we have to consider two botanical issues:
- How do plants get names and what kinds of names do they have?
- How are plants categorised? (The question of taxonomy).
And, as you might guess, things get a bit more complicated, because there are overlaps in these two questions.
- Plants have scientific names which are in Latin. They also have common names which can be in any language.
Scientific names have rules. They are binomial (consist of two parts) following a system that was first (formally) introduced by Carl Linnaeus. The first part of the name is the name of genus; the second part of the name refers to the specific species inside the genus.
E.g. Capsicum anuum is a species that belongs to the genus Capsicum.
Plants (as well as other living organisms) are organised in hierarchical groups: species < genus < family < order < ...< kingdom. There is a sub-group within the same species: variety. One species (with the same Latin name) can have many varieties, and their names are distinguished by adding 'var.'. E.g:
Capsicum annuum var. annuum and Capsicum annuum var. glabrisculum.
The source of confusion
are of course the common names which don't have such strict rules to follow. In general, in one language the same common name can refer to two (or more) different species (e.g. in different areas, common names often vary locally), two different common names can refer to the same species; a common name can refer to a whole genus; sometimes there are common names for a whole plant family (usually a common name of a plant that belongs to this family is used for this purpose).
Solanum tuberosum - common name: potato
Solanum lycopersicum - common name: tomato
These are two species that belong to the same genus: Solanum. They also belong to the same family: Solanaceae.
Another plant that belongs to this family?
Capsicum annuum - common name: pepper
Here - pepper is used as a common name for both the species Capsicum annuum and the genus Capsicum. This includes plants such as Capsicum baccatum and many others.
So, yes, you can say: Capsicum => pepper
They, together with potatoes, tomatoes and many other plants belong to the
Solanaceae family = Nightshade family.
There are several homonyms in English language that are spelled and pronounced pepper one of which is Pipper nigrum, a completely different spcies, of different genus, family and of course culinary uses.