What is briefly the main difference between a Siddha, a Sannyasi, a Bhakta and a Rishi?
In order to know the difference let's add few more terms to your list like Acarya, Muni and Upādhyaya:
Ācārya comes from the word Acharana, meaning to “implement” and “abide by”. An Ācārya can be a guru and can also be a jñani. However, the main difference is that an Ācārya himself/herself takes up an ashram (from among the four ashrams) and implements and abides by its karmā so as to become a living example in performing the karmā associated with that ashram. A good example is Śrī Śankaracharya, who was a jñani at such a young age, took up Sanyas:ashram and continued to retain in that ashram, implementing all its respective karmā. By doing so, he stood as an example – paving the way for future generations. He also took in shishya under him to pass on the tatva, and hence became a guru. Another good example is Dronācārya, who took up Gruhast:ashram and established an institution to teach all the Kauravas as students (note, the Pandavas are also Kauravas), and hence took the position as a guru.
Ācārya (आचार्य) refers to the “Śaiva preceptor”, whereas śiṣya refers to the “disciple”. The preceptor is of three types: Preraka,Bodhaka,Muktida ~Nigamajñāna Śaivāgamaparibhāṣāmañjarī (Wisdom Library)
An Upādhyaya or Adhyapaka (upa = position) is one who interprets the concepts of a scholar or Śāstra, becoming an intermediate towards explaining the concept. Hence Upādhyaya can be a profession, whereas Guru, or Jñani, or Ācārya are not professions. An Upādhyaya is mostly a position or a role in an academic institution (like a vice principal) and Adhyapaka is a teacher. Enlightenment is not a course nor a syllabus that one can complete through a trainer or an instructor.
Upādhyāya (उपाध्याय) refers to the “preceptor” and represents one of the ten persons suitable for rendering services ~~chapter 1.1 ādīśvara-caritra
Please note that a Sanyasi (Saṃnyāsi) is different from a Guru, although this is a common misinterpretation. Not every person who wears Kaashayam (orange or saffron-colored cloth) is a sanyasi and not every sanyasi is a Guru. Sanyasis are those who have moved beyond the three ashrams to unify with Iśvara through Vairagya in order to attain liberation (Mukti or mokṣa); hence one should never seek a sanyasi with desires or wishes. Now that we have had a small glimpse into the understanding of a Guru and Ḍharma, we must realize that Ḍharma and Guru constitute the quintessential concept of Sanātana Ḍharma Siddhānta, making it neither a religion nor a philosophy. Hence Sanātana Ḍharma should not be compared with any other religion or cult.
One in the renounced order of life, which is the highest order in the varṇāśrama-dharma system ~Śrīmad-Bhagavad-Gītā
A Rishi (rśi/Maharśi) (the closest word is a sage), is one who is not just a devotee or saint; he goes through tapasya (a deep meditative state) and becomes a medium to Iśvara and the ultimate truth. It is through a rishi that this truth is shared as Shruti and Kalpa(m). There are many classification like Maharśi, Brahmārśi, Rajarśi and more.
Ṛṣi (ऋषि).— "Ṛ" means "sound" which is the root word. So they are those who gave the sound of Vedas. ~Agni Purāṇa, Chapter 348
A Muni is an ascetic with austerities and lives on a resolve with devotion. Some remain in silence and some dwell in forever singing like Nāradā Mahamuni. Nāradā is also called Devarśi.
मुनीनामप्यहं व्यासः "munīnāmapyahaṃ vyāsaḥ" ~~Śrīmad-Bhagavad-Gītā.1.37
Both Rishis and Muni can enter into Gruhast (Married life and bear children).
A Yogi is one who has enhanced one’s physical body, the mental and psychological faculties, and the life energies that are in accordance with the energies of nature and the universe, using the discipline and the concepts of yoga.
"He rests and relaxes in His own atman." ~Śivapurāṇa 2.2.10.
A Siddha is one who has attained and excelled in a set of unique abilities (called Siddhis) that are not in the realm of a normal being, and are achieved through various procedures, austerities, yoga, tapasya, and more. There are various siddhis defined by Śāstra; however, eight are more prominently described in Yoga Sutras:
Anima, Mahima, Garima, Laghima, Prapti, Prakamya, Ishitva, and Vashitva ~Yoga Sutras Sage Kapila is defined as a Siddha, ~Śrīmad Bhagavād Gita
Hanuman is also defined to have excelled in all eight Siddhis. ~Ramayana
An Archaka is a person who performs upāchara (services) inside an inner sanctum (garbhalaya) of the deity in a temple. Archakas go through various sadhanas to consecrate themselves so that they are physically in a condition to serve a consecrate deity with an yantra (energy device). A Purohit, on the other hand, is a person who performs various rituals at the residence of people like pūjā, Kalpa, yajña, homa, havan, and more on various occasions like religious events or personal events like Gruhapravesha, Vastu pūjā, Namakarana, and more. A Pandit is one who has ripened in a specific scripture or sect and achieved mastery.
honoring of Diety, worshipping ~Manu-smṛti 11-224
Bhakta: is one who is devoted to one or more iconified Vedic Divinities.
Attached or devoted to "मन्मना भव मद्भक्तो मद्याजी मां नमस्कुरु" (manmanā bhava madbhakto madyājī māṃ namaskuru) Śrīmad-Bhagavad-Gītā.9.34