1. Love with a girlfriend, wife, fiancée, boyfriend, or husband, and which normally includes an implicit sexual component (this type of love is also known as "Eros")
2. Love for a relative, such as one's own child (this type of love is also known as "Storge")
3. LOVE, in the context of skepticism, which is an acronym that I created meaning "Logical, Objective, Verifiable Evidence"
   See also:  https://www.atheistfrontier.com/glossary/love.pl
4. Love of a pursuit, such as having a love for one's chosen line of work or for a hobby
5. Love for a friend (this type of love is also known as "Philia")
6. Love for nature, the universe, and everything (this type of love is also known as "Agape")
7. Love in a playful and uncommitted manner, which may include promiscuity (this type of love is also known as "Ludus")
8. Love rooted in a practical duty to procreate (this type of love is also known as "Pragma")
9. Love of self, and which may not necessarily include autosexuality; it is a somewhat common belief that this type of love is a pre-requisite for all other types of love (this type of love is also known as "Philautia")

Type 1 is, I assume, the most common one that comes to mind whenever the topic of love is introduced.

Type 2, along with most of the others, can be confusing for people who haven't differentiated between different types of "love."  For example, in consideration of type 2 the parent of a mass murderer may abhor what their child is doing time in prison for, but can also still love them as their child.  Many people have a hard time understanding this, but I suspect that the parent who visits their child in prison does understand.

The other types should be self-explanatory.

I don't agree that loving others requires loving oneself first since love is an emotion that doesn't have to be reciprocal (although reciprocity certainly makes it more meaningful).