Are the terms secular, agnostic, atheist applied by Indian people themselves to characterize existing attitudes in today's Indian Hindu society?
Added: Due to comment of @Whirl Mind I restrict my question to India's Hindu society.
The terms you use are modern Western terms but the various philosophical schools they represent have existed for thousands and thousands of years in India. Many of the works of those schools have few adherents nowadays as the predominant school for many centuries has been the Uttara Mimamsa, or Vedanta, school.
In his Introduction to his translation of the Brahma Sutras According to Sri Sankara, Swami Vireswarananda says (p ii):
...Traces of opposition against the religion of the Vedas are found in the Vedas themselves. This tidal wave of rationalism in its extreme form gave rise to such schools of thought as the Carvakas, which are extremely materialistic and anti-religious.
In the age immediately preceding Buddha and during his lifetime there was a great religious and philosophical upheaval in India. From the Bramajala-Sutras we learn that in his time there were as many as sixty-two different schools of philosophy in India. We also learn from Buddhistic literature the names of a good number of these teachers who were venerated in Aryavarta at the time--like Purana Kasyapa, Katyayana, Makkali Gosala, Nigantha Nathaputra the founder of Jainism and others. While these great souls represented Indian culture from an anti-Vedic standpoint there were many great names that represented the culture from the traditional standpoint--names that are still venerated by Hindu religion and culture.
The destructive criticism of everything in the old system by the Carvakas and others set the orthodox section to organize their belief on a more ratinalistic basis and render it immune against all such criticism. This led to the foundation of the six systems of orthodox Hindu philosophy--orthodox [footnote: Astika (orthodox) and Nasika (hetrodox) has nothing to go with belief or non-belief in the existence of a God. Samkhya and Mimamsa which did not accept an Iswara were yet regarded Astika (orthodox).] in the sense that they accepted the authority of the Vedas in things transcendental--while there were others who did not accept this authority and therefore were dubbed hetrodox, though otherwise they were the outcome of Upanisadic thought. The acceptance of the authority of the Vedas by these orthodox schools, however, does not mean that they accepted them in toto. Their allegiance to the Vedas varied widely and often it was too loose. Of the six orthodox schools viz Nyaya, Vaisesika, Samkhya, Yoga, Purva Mimamsa and Uttara Mimamsa or Vedanta, the last two are inimately connected with the Vedas, which is one of the reasons why they are not mentioned in the Jaina and Buddhistic literature, while the others are mentioned.
Literature of these different schools still exist and the arguments of the Vedantists (the overwhelming school of today) against these schools of thought still exist and are studied by scholars in India today. Many practicing Hindus may not be aware of this, but how many practicing Christians are aware of works like Dark Night of the Soul, or The Gnostic Gospels, The Philokalia, The Imitation of Christ, or Plotinus's neoPlatonism in The Six Enneads?
The Upanshads are not agnostic. The Upanishads do one thing - explain Brahman. Modern Vedanta, whether the dualistic school, the qualified non-dualists, or the non-dualists, are all founded on the Upanishads, the Brahma-Sutras, and the Gita. The second opening verse of the Brhama-Sutras (1.1.2) says (Sankara version):
(Brahman is that omniscient, omnipotent cause) from which proceeds the origin etc., (i.e. sustenance and dissolution) of this world.
And Ramanuja's version:
(Brahman is that Omniscient, Omnipotent, all merciful Being) from whom proceed the origin etc. (i.e. origin, sustenance and dissolution) of this (varied and wonderfully fashioned world).
Agnostic in the sense you are using it has nothing to do with the 'gods' of the Vedas. The proper term is devas, which is loosely translated as 'gods' but a more exact translation is 'shining ones'. The gods of the Vedas are positions, like chief minister or governor, held by jivas (bound souls), who have done certain karmic acts to gain the position. The positions are vacated at the end of a cycle to new souls for the next cycle. Those who held the positions formerly are then reborn to work out their existing karma. They are not 'gods' in the Western concept.
One can do ritualistic worship of devas to gain particular results in this world. But there is a sharp distinction between the worship of devas and the worship of Brahman. Swami Vivekananda says (Complete Works, V3, pp 59-60, and available here under the section Bhakti Yoga, sub-section Worship of Substitutes and Images - http://cwsv.belurmath.org/volume_3/vol_3_frame.htm :
The next points to be considered are the worship of Pratikas or of things more or less satisfactory as substitutes for God, and the worship of Pratimâs or images. What is the worship of God through a Pratika? It is — "Joining the mind with devotion to that which is not Brahman, taking it to be Brahman" — says Bhagavân Râmânuja. "Worship the mind as Brahman this is internal; and the Âkâsha as Brahman, this is with regard to the Devas", says Shankara. The mind is an internal Pratika, the Akasha is an external one, and both have to be worshipped as substitutes of God. He continues, "Similarly — 'the Sun is Brahman, this is the command', 'He who worships Name as Brahman' — in all such passages the doubt arises as to the worship of Pratikas." The word Pratika means going towards; and worshipping a Pratika is worshipping something as a substitute which is, in some one or more respects, like Brahman more and more, but is not Brahman. Along with the Pratikas mentioned in the Shrutis there are various others to be found in the Purânas and the Tantras. In this kind of Pratika-worship may be included all the various forms of Pitri-worship and Deva-worship.
Now worshipping Ishvara and Him alone is Bhakti; the worship of anything else — Deva, or Pitri, or any other being — cannot be Bhakti. The various kinds of worship of the various Devas are all to be included in ritualistic Karma, which gives to the worshipper only a particular result in the form of some celestial enjoyment, but can neither give rise to Bhakti nor lead to Mukti. One thing, therefore, has to be carefully borne in mind. If, as it may happen in some cases, the highly philosophic ideal, the supreme Brahman, is dragged down by Pratika-worship to the level of the Pratika, and the Pratika itself is taken to be the Atman of the worshipper or his Antaryâmin (Inner Ruler), the worshipper gets entirely misled, as no Pratika can really be the Atman of the worshipper.
But where Brahman Himself is the object of worship, and the Pratika stands only as a substitute or a suggestion thereof, that is to say, where, through the Pratika the omnipresent Brahman is worshipped — the Pratika itself being idealised into the cause of all, Brahman — the worship is positively beneficial; nay, it is absolutely necessary for all mankind until they have all got beyond the primary or preparatory state of the mind in regard to worship. When, therefore, any gods or other beings are worshipped in and for themselves, such worship is only a ritualistic Karma; and as a Vidyâ (science) it gives us only the fruit belonging to that particular Vidya; but when the Devas or any other beings are looked upon as Brahman and worshipped, the result obtained is the same as by the worshipping of Ishvara. This explains how, in many cases, both in the Shrutis and the Smritis, a god, or a sage, or some other extraordinary being is taken up and lifted, as it were, out of his own nature and idealised into Brahman, and is then worshipped. Says the Advaitin, "Is not everything Brahman when the name and the form have been removed from it?" "Is not He, the Lord, the innermost Self of every one?" says the Vishishtâdvaitin. — "The fruition of even the worship of Adityas etc. Brahman Himself bestows, because He is the Ruler of all." Says Shankara in his Brahma-Sutra-Bhâsya — "Here in this way does Brahman become the object of worship, because He, as Brahman, is superimposed on the Pratikas, just as Vishnu etc. are superimposed upon images etc."
The same ideas apply to the worship of the Pratimas as to that of the Pratikas; that is to say, if the image stands for a god or a saint, the worship is not the result of Bhakti, and does not lead lo liberation; but if it stands for the one God, the worship thereof will bring both Bhakti and Mukti. Of the principal religions of the world we see Vedantism, Buddhism, and certain forms of Christianity freely using images; only two religions, Mohammedanism and Protestantism, refuse such help. Yet the Mohammedans use the grave of their saints and martyrs almost in the place of images; and the Protestants, in rejecting all concrete helps to religion, are drifting away every year farther and farther from spirituality till at present there is scarcely any difference between the advanced Protestants and the followers of August Comte, or agnostics who preach ethics alone. Again, in Christianity and Mohammedanism whatever exists of image worship is made to fall under that category in which the Pratika or the Pratima is worshipped in itself, but not as a "help to the vision" (Drishtisaukaryam) of God; therefore it is at best only of the nature of ritualistic Karmas and cannot produce either Bhakti or Mukti. In this form of image-worship, the allegiance of the soul is given to other things than Ishvara, and, therefore, such use of images, or graves, or temples, or tombs, is real idolatry; it is in itself neither sinful nor wicked — it is a rite — a Karma, and worshippers must and will get the fruit thereof.