Nature of Craving in Charm and Impressing Others I noticed recently that I tend to try and impress others and gather attention, especially with women (I'm a man). I see this as problematic because unlike other sources of craving and attachment, its hard to avoid others. 

Thus, I'd like to understand the nature of the craving/grasping/etc. process in relation to trying to impress. What type(s) of attachment(s) are involved? How can one best relinquish a need to appear a certain way to others?

Also, I wonder: Is there any equivalent to 'overcompensating' in Buddhism? Overcompensating here means specifically attempting to augment the self because the self is in reality lacking, insecure. How does no-self affect this?

Thank you
What type(s) of attachment(s) are involved?

This is carving arising due to self-view.
There are 18 internal and external views across past, present and future making a total of 108 views of self rooted in carving.
18 Internal Preoccupations with Craving

(1) “I am”
(2) “I am this [I am like this]”
(3) “Thus am I [I am like that]”
(4) “I am otherwise”
(5) “I do not exist”
(6) “I exist”
(7) “May I be”
(8) “May I be this [be like this]”
(9) “May I be so [be like that]”
(10) “May I otherwise”
(11) “I might be”
(12) “I might be this [be like this]”
(13) “I might be that [be like that]”
(14) “I might be otherwise”
(15) “I shall be”
(16) “I shall be this [be like this]”
(17) “I shall be so [be like that]”
(18) “I shall be otherwise”

Tanhā Jālinī Sutta / (Vicarita) Tanha Sutta
18 External Preoccupations with Craving

(1) by this (form, etc), “I am”
(2) by this, “I am this [I am like this]”
(3) by this, “Thus am I [I am like that]”
(4) by this, “I am otherwise”
(5) by this, “I do not exist”
(6) by this, “I exist”
(7) by this, “May I be”
(8) by this, “May I be this [be like this]”
(9) by this, “May I be so [be like that]”
(10) by this, “May I otherwise”
(11) by this, “I might be”
(12) by this, “I might be this [be like this]”
(13) by this, “I might be that”
(14) by this, “I might be otherwise”
(15) by this, “I shall be”
(16) by this, “I shall be this”
(17) by this, “I shall be so”
(18) by this, “I shall be otherwise”

Tanhā Jālinī Sutta / (Vicarita) Tanha Sutta
With regard to external preoccupation with every interaction, one reinforces the self-view with further escalate to sensation resulting in the craving.

&quot;by this experience or interaction, I look good and I appear positively&quot;
&quot;by this experience or interaction, I look good but I appear negatively&quot;
&quot;by this experience or interaction, I look good but I appear neutral&quot;
&quot;by this experience or interaction, I look bad and I appear positively&quot;,
&quot;by this experience or interaction, I look bad but I appear negatively&quot;
&quot;by this experience or interaction, I look bad but I appear neutral&quot;
&quot;by this experience or interaction, I look neutral and I appear positively&quot;
&quot;by this experience or interaction, I look neutral but I appear negatively&quot;
&quot;by this experience or interaction, I look neutral but I appear neutral&quot;

More the one appears positive with respect to the self-evaluation and the perception one appear, one experiences deeper shades of pleasantness. Similarly with it is negative one experiences unpleasantness. When there is no disparity, i.e., one's evaluation of oneself matches one's expectation of how one is perceived by others, there is a neutral sensation.
This is further appplified based on the people one tries to project the evaluation, the level one want to make an impression, the number of people involved as well as the source of the experience. The grouping of people could be:

oneself
special people (friends, foes, coworkers and family)
acquainted people
strangers

If the source is a close person then the feeling is more. If a loved one does something unexpected one feels more strongly than a stranger. E.g. if someone you admire is negative about you the pain is more.
If something about you is known by many vs few it will increase the feeling. E.g. if someone does something improper and is caught one may try to stop the knowledge of it spreading as one sees this as damaging.
If one it trying hard to impreses then this also will intensify the sensations. E.g. If one does badly at school it might be harder to let your parents know the results than a stranger who does not care as much.
The combinations give different shares of feelings:

something done by a special person, which affects how I want to project my self, which is projected to other special people
something done by a special person, which does not affects how I want to project my self, which is projected to other special people
something done by a special person, which affects how I want to project my self, which is projected to other people who I care about
something done by a special person, which affects how I want to project my self, which is projected to other people who are strangers

...

How can one best relinquish a need to appear a certain way to others?

This is by eliminating the classes of craving leading to self-view.