No, for Arms of Hadar, you are not counted as within it's area of effect due to an exception the spell creates
The general rules for Range state:
Spells that create cones or lines of effect that originate from you also have a range of self, indicating that the origin point of the spell's effect must be you [...].
The rules for Area of Effect state:
A spells description specifies it's area of effect, which typically has one of five different shapes [...]. Every area of effect has a point of origin, a location from which the spell's energy erupts. [...] Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or object.
A spell's effect expands in straight lines from the point of origin.
All of the five common areas of effect specify whether or not it's point of origin is included in the area of effect. Arms of Hadar uses a non-standard Range/Area of
Self (10 foot radius)
which is not a sphere or a cylinder as those shape modifiers are not specified. Under normal circumstances, this would leave it up to a DMs adjudication, and based on the above quoted rules, the default, as written, is that point's of origin are included in the spells area of effect.
That being said, the rules also state, explicitly, that the area of effect is specified in the spell's description, and in this case, the description points us in the correct direction (emphasis mine):
Tendrils of dark energy erupt from you and batter all creatures within 10 feet of you.
By this description, and in particular the phrase "erupt from you" creates an exception that excludes you, the caster, from the area of effect. This is because the verb erupt specifies an outward direction under it's normal English interpretation (eg a volcando erupting lava means the volcano violently pushed lava out and away from it).
Since you are not included in the spells area of effect, you are then excluded from the remainder of the spells description (and in particular the damaging component of the spell).