Teaching or giving advice without mentioning the Buddha

It was never the Buddha's wish to promote himself, of course, and he never required that he be credited with the teachings he taught. In the Mahaparinibbana Sutta, he complains of them making him out to be their leader:

32. Thus spoke the Venerable Ananda, but the Blessed One answered him, saying: "What more does the community of bhikkhus expect from me, Ananda? I have set forth the Dhamma without making any distinction of esoteric and exoteric doctrine; there is nothing, Ananda, with regard to the teachings that the Tathagata holds to the last with the closed fist of a teacher who keeps some things back. Whosoever may think that it is he who should lead the community of bhikkhus, or that the community depends upon him, it is such a one that would have to give last instructions respecting them. But, Ananda, the Tathagata has no such idea as that it is he who should lead the community of bhikkhus, or that the community depends upon him. So what instructions should he have to give respecting the community of bhikkhus?

-- DN 16 (Vajjira, trans)

And in the Udumbarika-Sīhanāda Sutta, he expresses his intentions most eloquently:

‘But, Nigrodha, I tell you this: Let an intelligent man come to me who is sincere, honest and straightforward, and I will instruct him, I will teach him Dhamma. If he practises what he is taught, then within seven years he will attain in this life to that unequalled holy life and goal, for the sake of which young men of good family go forth from the household life into homelessness, by his own knowledge and realisation, and he will abide therein. Let alone seven years — in six years, five, four, three, two years, one year, ... seven months, six months, five, four, three, two months, one month, half a month. Let alone half a month — in seven days he can gain that goal.

23. ‘Nigrodha, you may think: “The ascetic Gotama says this in order to get disciples.” But you should not regard it like that. Let him who is your teacher remain your teacher. Or you may think: “He wants us to abandon our rules.” But you should not regard it like that. Let your rules remain as they are. Or you may think: “He wants us to abandon our way of life.” But you should not regard it like that. Let your way of life remain as it was. Or you may think: “He wants to establish us in the doing of things that according to our teaching are wrong, and are so considered among us.” But you should not regard it like that. Let those things you consider wrong continue to be so considered. Or you may think: “He wants to draw us away from things that according to our teaching are good, and are so considered among us.” But you should not regard it like that. Let whatever you consider right continue to be so considered. Nigrodha, I do not speak for any of these reasons ...

‘There are, Nigrodha, unwholesome things that have not been abandoned, tainted, conducive to rebirth, fearful, productive of painful results in the future, associated with birth, decay and death. It is for the abandonment of these things that I teach Dhamma. If you practise accordingly, these tainted things will be abandoned, and the things that make for purification will develop and grow, and you will all attain to and dwell, in this very life, by your own insight and realisation, in the fullness of perfected wisdom.’

24. At these words the wanderers sat silent and upset, their shoulders drooped, they hung their heads and sat there downcast and bewildered, so possessed were their minds by Māra. Then the Lord said: ‘Every one of these foolish men is possessed by the evil one, so that not a single one of them thinks: “Let us now follow the holy life proclaimed by the ascetic Gotama, that we may learn it — for what do seven days matter?”’

-- DN 25 (Walshe, trans)

So, certainly, there seems no need to mention the Buddha when sharing the teachings. The big proviso, though, is that it is an egregious evil to pretend that it came from another source, especially that you yourself came up with the teachings. Like plagiarism, misappropriation is a terrible thing, especially in regards to such profound teachings.