Though my practice is "Mahayana" (in quotes), I have a great and abiding respect for the Theravada school and consider (as many do) the term "Hinayana" (Sanskrit for Smaller Vehicle) disrespectful of that tradition. My question is, using the term "Mahayana" (Sanskrit for Great Vehicle) has a built-in comparison and seeming insult to that other tradition. Therefore, is there another term for "Mahayana" that can be used to discuss that remarkable tradition without bringing in that un-needed and negative comparison to another school? Thanks, greatly.
As a fellow 'Mahayana' practitioner, I share your great respect for Theravadin buddhism and appreciate your sensitivity to the polemical origins of the term 'Mahayana' (I've used it many times myself, hopefully without causing offense; but try not to ever use the term 'Hinayana'). Because the term 'Mahayana'covers an extraordinary diversity of paths, you may want to just name your specific tradition if the implication of this general term troubles you.
Another possible approach: 'bodhisattva-yana' is sometimes suggested as a general substitute term. In historical terms (according to Buswell's Encyclopedia of Buddhism), it apparently refers to a development in India that preceded Mahayana. However, it seems that for practical purposes, referring to yourself as a follower of the bodhisattva way would give fellow practitioners in any school of buddhism enough context to understand where you were coming from, without any derogatory implications.
Hīnayāna means "a defective vehicle". Hīna does not mean "lesser". The reason we think it might is because of Kumārajīva's influential Chinese translation of the Saddharmapuṇḍarikā Sūtra which translates hīna as 小 xiǎo 'small'. But this is misleading.
For example in the Saddharmapuṇḍarikā Sūtra we find the following stanza:
ekaṁ hi kāryaṁ dvitiyaṁ na vidyate
na hīnayānena nayanti buddhāḥ ||55||
There is only one method, not a second,
The Buddhas do not lead by a defective way.
Give that this text is often cited as a locus classicus for prejudice against the so-called Hīnayāna, a supposedly lesser Buddhist teaching, this verse shows that it might need to be reconsidered. The Buddha's do not teach defective ways to awakening.
Part of the reason the pejorative sense was lost was that Buddhists outside India stopped using Sanskrit and didn't really understand the history of the Sanskrit words they retained.
Up to about the 4th century one can use the term "Mainstream" to refer to Buddhism that is not Mahāyāna. Around this time Mahāyāna itself was becoming the Mainstream in India (for the first time). Indeed in medieval India non-Mahāyāna Buddhism is almost irrelevant, except that all ordinations are still done according to Vinaya rules that stretch back to early Buddhism (which is still true of Vinaya ordinations). Perhaps the first non-Vinaya ordination was conducted in Japan in 822 CE by Saichō, founder of the Tendai School. Far more relevant in Indian at this time are distinctions such as Madhyamaka and Yogācāra.
The sectarian divides of India were not necessarily reproduced outside India. The Mahāyāna/Hīnayāna distinction never really meant anything in Tibet for example because there never were any Buddhists who might have worn the latter label. Tibetans use the word entirely differently. China did at least receive a substantial number of Mainstream Buddhist texts, but not until the very end of the Mainstream period (ca. 4th-5th century). But from the first Mahāyāna texts were important in China.
So the terms to use depends to some extent whether one is referring to ordination or to doctrinal differences. And what time period and place one is referring to.
In the present day there is only one school that might have been traditionally associated with Mainstream Buddhism (by this definition). This is Theravāda. They are not part of a broader category on any doctrinal basis and mostly based on early medieval interpretations of early Buddhist teachings, so not Mainstream doctrinally anyway. So the term Theravāda is adequate for most purposes. However they share with Vinaya monks everywhere the same kind of ordinations (albeit with substantial differences from sect to sect).
But I see no reason not to use the term Mahāyāna. It's a bit vague, considering how much internal disputing goes on between Mahāyāna sects, but it is the right word to use. Though modern scholarship has substantially changed the view of Mahāyāna recently.