So if siRNA appears to be just a subtype of miRNA, can we confirm
that? Or are there any other differences that make these two molecules
different kinds of small non-coding RNAs?
siRNA and miRNA are not identical, but similar. It's not incorrect to consider them similar phenomena in biology, but history places them apart and there exist small differences. Both are processed inside the cell by the enzyme Dicer and incorporated into the RISC complex where their target is degraded ("RNA interference"). However, there are important differences, in case you want to be absolutely correct.
siRNA is typically exogenous (introduced from outside by viruses or by engineering). It typically has 100% complimentary to its target mRNA, because we want to reduce off-target binding. If designed well, they target only the intended target and are checked to be incompatible with any other potential mRNA targets, i.e. are designed to be specific.
miRNA is endogenous (occurs naturally within cells) and is single stranded in vivo. Its complimentary is not always 100% perfect, and often has multiple targets. This is the major difference between the two. It pairs with its target mRNA imperfectly; it simply inhibits translation (protein production from the mRNA). In rare cases where it's a perfect match, it causes the cleavage and destruction of the target mRNA just like siRNA. If I remember correctly, plants miRNAs, in contrast to animal miRNAs, usually have perfect complementarity with their targets. miRNAs are also derived from shorter stemloop RNA products than siRNA.
As you probably know, both are non-coding RNAs and do not enter ribosomes for translation. They have similar effects on gene expression. If you are interested in the very first reports of either kinds of RNA, here goes:
miRNAs discovered in 1993 in worms.
siRNAs discovered in 1999 in plants.
Notice the phrase antisense RNA in both titles. Interesting!
Further reading, in case you are interested.
Why does a cell need siRNA?
This is a very different question. Why? Probably to fine-tune the expression of its genes. Small RNA species can accumulate, aggregate, and regulate mRNA levels in ways that are probably not possible inside the nucleus at the DNA transcription level. Small RNAs are also understood to be epigenetic and function in the cytoplasm and endoplasmic reticulum!