The link referred to uses the ideas of the monk Yuttadhammo as a guide. Yuttadhammo is connected to a certain idiosyncratic Burmese meditation tradition that appears to misapprehend Anapanasati.
The very fact that the Pali scriptures report Anapanasati is 'The Tathagata's Dwelling' (SN 54.11) shows believing Anapanasati to be only 'samatha' is an error.
If anyone, bhikkhus, speaking rightly could say of anything: "It is a noble dwelling, a divine dwelling, the Tathagata's dwelling", it is of
concentration by mindfulness with breathing that one could rightly say
SN 54.11 (no direct link)
Anapanasati, when performed correctly, develops both samatha (tranquility) & insight (vipassana) together. In the majority of discourses in the Pali suttas, Anapanasati is recommended as a complete system of meditation, where samatha (tranquility) & insight (vipassana) are developed in tandem.
There cannot really be different kinds of meditation, when meditation is practised correctly & successfully. The reason for this is because when attention & concentration are properly developed, insight will naturally occur as an automatic result. But if concentration is practised incorrectly, such as by suppressing & forcing the mind, insight will not occur and the samatha that occurs will be unclear & drowsy.
Similarly, if the Burmese 'insight' techniques can be practised successfully, true insight will lead to the mind's hindrances/defilements/thoughts stopping, which will result in the mind automatically developing concentration.
This is why the Dhammapada states there can be no concentration without insight & no insight without concentration.
372. There is no meditative concentration for him who lacks insight and no insight for him who lacks meditative concentration. He in whom
are found both meditative concentration and insight, indeed, is close
The meditation methods developed in Burma, which Yuttadhammo teaches, appear to have been developed to avoid the lethargy of mind that occurs from when samatha meditation is developed improperly, which is extremely common. It is probably a fact that most people do not practise Anapanasati in an ideal manner, which results in misunderstandings about Anapanasati.
The very fact that most translators translate 'Anapanasati' as 'Mindfulness Of Breathing' demonstrates an inherent misunderstanding of what 'Anapanasati' is (since the translation should be 'Mindfulness With/When/While Breathing').
The so-called 'Insight' techniques taught by Yuttadhammo are designed to keep the mind open, awake & alert; to avoid the lethargy of wrong anapanasati practise.
While this may seem to be a very positive thing, the best thing is to correct one's misunderstandings of Anapanasati rather than preach misunderstandings about Anapanasati.
Ultimately, we must choose the technique that is best for us. For those unable to practise Anapanasati as instructed in the suttas (which is based on the practise of 'letting go' called 'vossagga'), the Burmese 'insight' methods of Mahasi Sayadaw that Yuttadhammo teaches may be a better option if they wish to develop alertness & attention.
...a monk develops mindfulness as a factor for awakening
dependent on seclusion, dependent on dispassion, dependent on
cessation, resulting in relinquishment (vossagga).
He develops analysis of
dhammas (insight) as a factor for awakening dependent on seclusion, dependent on
dispassion, dependent on cessation, resulting in relinquishment
And what is the faculty of concentration? There is the case where a monk, a disciple of the noble ones, making it his object to let go
(vossagga), attains concentration, attains singleness of mind.