Intention for ablution

Is it sufficient to make a wudu valid according the shafi'i
madhhab by intending "nawaytul-wudu liLlahi ta''Ala" (I intend to perform ablution for Allah, most High) when first washing the face?

In the name of Allah, the most Merciful, the most Compassionate.

It is sufficient to intend ablution (Wudu), mentioning for instance, "nawaytu al-Wudu", but it is better to intend the obligation of ablution (Fard al-Wudu). Shaykh Ibn Hajar al-Haytami says in Tuhfa that it is sufficient to intend the obligation of ablution (Fard al-Wudu) or just ablution, as you mentioned. However, Ba 'Ashin mentions in Bushra al-Karim that it is better to intend the obligation of ablution (Fard al-Wudu) to avoid the difference of opinion with those who say this is obligatory.

One makes this intention at the time of washing the first portion of the face. However one must make an intention before this in order to be rewarded for the sunnas which precede the face. The intention for the sunnas should be made when beginning to wash the hands and it coincides with saying "Bismillah". Shaykh Ibn Hajar says in Tuhfah:

So one intends along with it [the saying of "Bismillah"] while washing the hands, since this is the meaning of its [ablution's] beginning in the primary text. [One does this] by making the intention coincide with saying "Bismillah" at the beginning of washing them. Just as it [the intention] coincides with the opening Allahu Akbar of prayer (Ibn Hajar al-Haytami, Hawashi al-Shirwani wa Ibn Qasim al-Abbadi 'Ala Tuhfah al-Muhtaj Bisharh al-Minhaj [Beruit, Dar Ihya' al-Turath al-'Arabi], 1:225).

It is best that the intention which is made along with washing the hands be merely to perform the sunnah, by saying, for instance "I intend the sunnas of ablution" (nawaytu sunan al-Wudu'), as Abdul-Hamid al-Shirwani conveyed from his shaykh al-Bajuri.

So here one is doing three things simultaneously: (1) intending the sunnas of ablution, (2) saying "Bismillah", (3) beginning to wash their hands. Abdul-Hamid al-Shirwani said, quoting his shaykh al-Bajuri:

So he joins in this action between his heart, tongue, and limbs. For he has used his heart for the intention, his tongue saying "Bismillah", and his limbs for washing at the same time (Abdul-Hamid al-Shirwani, Hawashi al-Shirwani Wa Ibn Qasim al-Abbadi 'Ala Tuhfah al-Muhtaj Bisharh al-Minhaj [Beruit, Dar Ihya' al-Turath al-'Arabi], 1:225).

It is praiseworthy to mention this intention silently with the tongue as well, but one is also saying "Bismillah". So when do they mention the intention? It is possible that one says it after saying "Bismillah" so the blessing of saying "Bismillah" encompasses it. Or it is possible that one mentions the intention prior to saying "Bismillah", and then makes the intention in their heart while saying "Bismillah". Just as they say the intention prior to the opening Allahu Akbar of the prayer. Shaykh Ibn Hajar mentioned both of these possibilities in Tuhfah, and in some of his other books he gave preference to the first. The first possibility is also what is in Nihaya and Mughni. And Allah knows best.

This is how one makes intention for ablution in a nutshell. We went into a lot of detail because what is mentioned in most books needs more commentary. And Allah alone gives success (tawfiq).