Majjhima Nikaya 2 goes through a list of methods to abandon craving, including the two you mention.

Two key questions for this are: what is the main deep underlying purpose behind your question? What do you expect to achieve after whatever "method" you start following?

If what you want is "simply" relax your head for awhile, I think any approach that takes such negative thoughts away in that very moment may be useful. But I suspect that won't solve the problem for good, because the root of those thought of ego and pain will come back eventually.

If what you want is to eliminate the root cause of those thoughts in order to cease any future arising of them, you may need to eliminate the condition that make them arise, and not just calm those thoughts for awhile.

Have you noticed that there are periods where those thoughts are not there, and instead there are other kinds of thoughts? Maybe ones without any special relevance of negetive effect on your feelings and emotions. Another good questions to ask yourself is: under what circumstances do such negative thought arise? What is the condition that, when present, makes them express? Is such condition/s always there, or it comes and goes randomly or every once in a while?

All of this comes reflected on the ideas expressed by the Buddha of Dependent Co-arising (the conditions that give rise to dissatisfaction, stress and mental suffering) and in his Path based on the development of wisdom (that changes one's way of processesing the world, by changing one's deepest underlying tendencies and assumptions about "oneself" and the world) and the cultivation of non-attachment (and release of those conditions and attachments that perpetuate the cycle of suffering).

Kind regards!

These are not mutually exclusive. In fact, the latter method contains the 1st method you mentioned as the last resort. So do both in tandem as needed:

OP: One would be to try to ignore the thoughts, not give them energy, not identify with them, be present and in time they would go into the background. Basically they are just thoughts, they are not you.

You can try "ignoring" thoughts by focusing on a particular object hence developing some level of Jhana to reduce mental chatter through the Jhana factors like initial and sustained application:

When, bhikshu, this samadhi has been cultivated, well cultivated by you, then you should train yourself thus:

“I will dwell exertive, clearly aware, mindful,

observing [contemplating] [body | feeling | mind | dhamma] the in the [body | feeling | mind | dhamma],

removing covetousness and displeasure [discontent] in regard to the world.”

Thus, bhikshu, you should train yourself.

When, bhikshu, this samadhi has been cultivated, well cultivated by you, then, you, bhikshu,

• THE 1 ST DHYANA:

should cultivate this samadhi with initial application, with sustained application;

should cultivate this samadhi without initial application, with only sustained application;

• THE 2 ND DHYANA:

should cultivate this samadhi without initial application, without sustained application;

should cultivate this samadhi with zest;

• THE 3RD DHYANA:

should cultivate this samadhi zest-free;

should cultivate this samadhi attended by comfort;

• THE 4 TH DHYANA:

should cultivate this samadhi attended by equanimity.

Saṅkhitta Dhamma Sutta

in the 2nd Jhana and beyond there is no mental chatter.

OP: Other approach would be to sit down, meditate and try to go deep inside to find the root of those thoughts (I suspect I have myself deep down :P). Try to see clearly the constant river of thoughts that is happening inside (things I notice is just a small surface of things).

Thought proliferation is due to the following process:

“Bhikshu, as regards the source from which proliferation of conception and perception assails a person: if one were to find nothing there to delight in, nothing there to welcome, nothing to cling to—this is the end of

• the latent tendency of lust,
• the latent tendency of aversion,
• the latent tendency of views,
• the latent tendency of doubt,
• the latent tendency of conceit,
• the latent tendency of desire for existence, and
• the latent tendency of ignorance.

This is the ending of the taking up of the rod and the sword, quarrels, disputes, mayhem [strife], slandering and lying —here these evil unwholesome states cease without remainder.”

…

Avuso, dependent on the [eye | ear | nose | tongue | body | mind] and [form | sound | smell | taste | touch | mind-object], [eye | ear | nose | tongue | body | mind]-consciousness arises.

• The meeting of the three is contact.

• With contact as condition, there is feeling.

• What one feels, one perceives.

• What one perceives, one thinks about.

• What one thinks about, one mentally proliferates.

From that as source, proliferation of conception and perception assails a person regarding past, future and present [forms | sounds | smells | tastes | touch | mind-objects] cognizable through the [eye | ear | nose | tongue | body | mind].

Madhu,piṇḍika Sutta

There 3 crucial points in this process:

1. feeling/sensation
2. one perceives then thinks and finally build views
3. mental chatter resulting in thought proliferation (papañca)

There are 3 things to be done at each juncture:

Feeling/sensation

Feeling/sensation result in perception (Madhu,piṇḍika Sutta above), craving (see Dependent Arising), 3 unwholesome roots (Pahāna Sutta). So one has to break the progression beyond feeling/sensation by:

i. On seeing a form with the eye,

• one investigates the form that is the basis for mental joy,
• one investigates the form that is the basis of mental pain,
• one investigates the form that is the basis of equanimity.

ii. On hearing a sound with the ear,

• one investigates the sound that is the basis for mental joy,
• one investigates the sound that is the basis of mental pain,
• one investigates the sound that the basis of equanimity.

iii. On smelling a smell with the nose,

• one investigates the smell that is the basis for mental joy,
• one investigates the smell that is the basis of mental pain,
• one investigates the smell that is the basis of equanimity.

iv. On tasting a taste with the tongue,

• one investigates the taste that is the basis for mental joy,
• one investigates the taste that is the basis of mental pain,
• one investigates the taste that is the basis of equanimity.

v. On feeling a touch with the body,

• one investigates the touch that is the basis for mental joy,
• one investigates the touch that is the basis of mental pain,
• one investigates the touch that is the basis of equanimity.

vi. On cognizing a mind-object with the mind,

• one investigates the mind-object that the basis of mental joy,
• one investigates the mind-object that is the basis of mental pain,
• one investigates the mind-object that is the basis of equanimity.

...

If he feels a pleasant feeling,

• he understands that it is impermanent;
• he understands that it is not to be clung to;
• he understands that there is no delight in it.

If he feels a painful feeling,

• he understands that it is impermanent;
• he understands that it is not to be clung to;
• he understands that there is no delight in it.

If he feels a neutral feeling,

• he understands that it is impermanent;
• he understands that it is not to be clung to;
• he understands that there is no delight in it.

If he feels a pleasant feeling, he feels it in a detached manner.

If he feels a painful feeling, he feels it in a detached manner.

If he feels a neutral feeling, he feels it in a detached manner.

Dhātu Vibhaṅga Sutta

To prevent your mental reaction to sensation becoming unwholesome you can try:

“Nothing is worth clinging to”

When this was said, the venerable Mahā Moggallāna said this to the Blessed One: “In what way, bhante, in brief, is a monk freed through the destruction of craving, that is, one who has reached total perfection, the total security from bondage, the total holy life, the total consummation, the highest amongst gods and humans?”

“Here, Moggallāna, the monk has learned that nothing is worth clinging to. And, Moggallāna, a monk has learned that nothing is worth clinging to, thus: he directly knows all things [he directly knows the nature of the all]. Having directly known the nature of all things, he fully understands all things.

Having fully understood all things, he knows whatever feelings there are, whether

• pleasant,

• painful or

• neither painful nor pleasant.

As regards to those feelings, [Section on Disillusionment and Revulsion (Nibbida) follows]

• he dwells contemplating impermanence in them;

• he dwells contemplating dispassion [fading away of lust] in them;

• he dwells contemplating ending (of suffering) in them;

• he dwells contemplating letting go (of defilements).

When he dwells

• contemplating impermanence in them,

• contemplating dispassion in them,

• contemplating ending in them,

• contemplating letting go,

he does not cling to anything in the world.

Not clinging, he is not agitated; being not agitated, he himself surely attains nirvana.

Pacalā Sutta

One Perceives then Thinks and Finally Build Views

At this point, cravings and unwholesome would have arisen which also gives rise to perversions. For a worldly person perceptions, thoughts, views that arise are subjected to the 4 perversions (vipallasa), which are:

• what is impermanent is taken to be permanent;
• what is painful is taken to be pleasurable;
• what is not self is taken to be a (or the) self; and
• what is impure [unattractive or repulsive] is taken to be pure [attractive or beautiful]

One should try to break this train also when feelings arise. The method to deal with preception is also as in the previous part, hence I will not repeat it here.

If

• perceptions,
• thoughts, or
• views

do arise one should straighten them by contemplating:

• Impermanent as impermanent
• Painful as painful
• Not-self as not-self
• Impure [unattractive or repulsive] as impure

See Vipallasa Sutta

Mental Chatter Resulting in Thought Proliferation

If one could not stop the progression of events at feeling/sensation, perception and thinking then this results in mental chatter which is through proliferation. This results in latent tendencies (Madhu,piṇḍika Sutta) and is the last point one can stop this train of events progressing further but by this time craving, unwholesome roots and perversion (vipallasa) of perception, thoughts and view would have already arisen.

Metal proliferation can be cut out thought Jhana as from the 2nd Jhana onwards there is no thinking and pondering (vitakka-vicāra).

According to Dantabhūmi Sutta one should abandon thoughts with regard to the Sathipattanas namely thoughts regarding or originating from the body, feeling, mind and dhammas:

(1) dwell observing the body in the body,

and do not think (any) thought regarding the body;

(2) dwell observing feelings in feelings,

and do not think (any) thought regarding feelings;

(3) dwell observing the mind in the mind,

and do not think (any) thought regarding the mind [thoughts];

(4) dwell observing dharmas in the dharmas,

and do not think (any) thought regarding dharmas [realities].

With the stilling of initial application and sustained application,

by gaining inner tranquillity and oneness of mind,

he enters and dwells in the second dhyana, free from initial application and sustained application, accompanied by zest and joy, born of concentration.

And with the fading away of zest,

he dwells equanimous. Mindful and clearly comprehending, he experiences joy with the body.

He enters and dwells in the third dhyana,

of which the noble ones declare, ‘Happily he dwells in equanimity and mindfulness.’

With the abandoning of joy and abandoning of pain,

and with the earlier disappearance of pleasure and displeasure,

he attains and dwells in the fourth dhyana that is neither painful nor pleasant,

and with mindfulness fully purified by equanimity.

This way latent tendencies (anusaya) and further vipallasa and sankhā through the papañca-saññā-sankhā process will not arise. The method of practice is given above and also found in the Saṅkhitta Dhamma Sutta quoted above.

Also one can use another object of mentation like Ānâpāna,sati Sutta.

Essentially this is bringing one's focus to a given object and repeatedly refocusing on the object again to sustain one's focus on the particular object so one is not distracted by unwholesome trains of thoughts. Also, to get into Jhana in the moment of need requires greater length and effort in pratice through.