Agni and Saumya are different names of the same tirtha found on the right hand only.
Here is what the Bhaviṣya Purāṇa Brāhma Parva Chapter 3 says:
पञ्चतीर्थानि विप्रस्य श्रूयन्ते दक्षिणे करे। देवतीर्थं पितृतीर्थं ब्रह्मतीर्थं च मानद॥६२॥
प्राजापत्यं तथा चान्यत्तथान्यत्सौम्यमुच्यते। अङ्गुष्ठमूलोत्तरतो येयं रेखा महीपते॥६३॥
ब्राह्मं तीर्थं वदन्त्येतद्वसिष्ठाद्या द्विजोत्तमाः।
कायं कनिष्ठिकामूले अङ्गुल्यग्रे तु दैवतम्॥६४॥
तर्जन्यङ्गुष्ठयोरन्तः पित्र्यं तीर्थमुदाहृतम्।
करमध्ये स्थितं सौम्यं प्रशस्तं देवकर्मणि॥६५॥
Oh King, five tirthas in the right hand of Brahmanas are heard off - (1)Devatirtha, (2)Pitrtirtha (3)Brahmatirtha, (4)Prajapatya and (5)Saumya. Beyond the base of the thumb is the Brahma tirtha as per Vasishtha and other foremost Brahamanas. At the base of the little finger is the Ka (Prajapatya), at the tip of the fingers the Daiva, between the thumb and index finger is called the pitrya. In the middle of the hand is the Saumya which is famous for Deva worship.
As per the above Vasistha and others have described the five tirthas of the hand, which obviously leads one to verify the kalpasutras by eminent brahmanas, the most important being Vasishtha. In this regard, the Vasiṣṭha Dharmasutra 3.67 mentions that the middle of the hand is called the Agni tirtha, thought the other tirthas are somewhat inconsistent with the majority of texts. The views of other eminent brahmanas are as follows:
- Pāraskara Pariśiṣṭa Śaucasutra says:
ब्राह्मणस्य दक्षिणहस्ते पञ्चतीर्थानि भवन्ति अङ्गुष्ठमूले ब्रह्मतीर्थे कनिष्ठिकाङ्गुलिमूले प्रजापतितीर्थे तर्जन्यङ्गुष्ठमध्यमूले पितृतीर्थमङ्गुल्यग्रे देवतीर्थे मध्येऽग्नितीर्थमित्येतानि तीर्थानि भवन्ति॥२॥
A Brahmana's right hand has the five tirthas. At the base of the thumb is the Brahmatirtha, at the base of the little finger the Prajapati tirtha, at the base of the middle portion between the thumb and index finger is the Pitr tirtha, at the tip of the fingers is the Deva tirtha and in the middle of the hand is the Agni tirtha.
- Bodhāyana Gṛhyaśeṣa Sutra 4.7.2 says:
ब्राह्मणस्य दक्षिणे हस्ते पञ्चतीर्थानि भवन्ति अङ्गुलीमध्ये देवतीर्थमङ्गुल्यग्र आर्षे तीर्थे मध्येऽग्नितीर्थमङ्गुष्ठतर्जन्योर्मध्ये पैतृकतीर्थमङ्गुष्ठतलेऽतिहृत्य पश्चाल्लेखं तद्ब्रह्मतीर्थम्॥
A Brahmana's right hand has the five tirthas. In the middle of the fingers is the devatirtha, at the tips the Rishi tirtha, in the middle of the hand is the Agni tirtha; between the thumb and the index finger the pitr tirtha and beyond the line at the base of the thumb is the Brahma tirtha
Here it can be seen that there is a slight inconsistency as regards the placement of the deva tirtha and rishi tirtha, but the Bodhāyana Dharmasutra 22.214.171.124-16 clarifies that the deva tirtha is at the tips and rishi tirtha at the base. The above seems to be a possible minor error in copying the manuscript. The exact position can be better understood from the Padma Purana (given below) and Agni Purana (referred to in the nitya puja prakash)
- The Vaikhānasa Smārtasutra Praśna 1 Khaṇḍa 5 reads:
On the right hand, the middle part of the palm is the part sacred to Agni....
and many more. About Saumya tirtha, I could only find one other reference from the Padma Purana where the Saumya tirtha aspect has been dealt with very clearly. It says that the middle part of the hand is sacred to Agni as well as Soma.
The part of the hand situated on the line at the root of the thumb is called Brāhma. The part between the thumb and the fore finger is said to be the holy place of the manes. The part below the little is called Prājāpatya. The tip of a finger is said to be divine. The same is called Ārṣa (‘of the sages’). The root is daiva or ārṣa, and the middle part is said to be Āgneya (of Agni). The same is sacred to Soma. - Padma Purana Svargakhaṇḍa, Chapter 52
Generally in rituals the right hand is preferred and as per the above scriptures too we do not see mention of the left and right hand having different tirthas, as against the Nitya Karma Puja Prakash. They have drawn reference from only the Agni Purana in that. Thus it is safe to assume, based on the majority view and especially the Padma Purana, that the Agni and Saumya tirtha are different names of the middle portion of the right hand only, sacred to both deities.
Relevant screenshot from Nitya Karma Puja Prakash: