Oh, no. There are several verb groups, and each groups has subgroups with different Present tense patterns. You will learn them as you go down the Hebrew tree. But since this is the first group (binyan) - pa'al, there are three distinct subgroups within this binyan.
The first one, as you mention has vav inserted between the first and second radical of the root, like אוכל or שותה. So, their pattern is xOxEx (x representing the letter of the root)
Second subgroup are verbs that in Present tense have two letters, such as שם or נח. Their pattern is xAx.
The third, and by far the smallest subgroup is a group of so called stative verbs, which have a pattern xAxEx, just like ישן (yashen - sleep) or גדל (gadel - grow) or some other that are considered adjectives in modern Hebrew, but were considered verbs in Biblical Hebrew - מלא (male - full), יפה (yafe - beautiful) and so on.
This is just pa'al. Other binyanim have other patterns.
Verbs in Hebrew are split into 7 different groups. The same 3 letters can be used in multiple groups.
I learn = אני לומד (Ani Lomed)
I teach = אני מלמד (Ani Melamed)
A vav as the second letter means it's either in group פעל (Pa'al) or הפעל (huf'al).