Let us start with the actual Ashtakshara Swayamvyakta temples which Sri Ramanujacharya listed:
Srirangam venkatadrischa srimushnam totaparvatam
salagramam pushkaramcha naranarayanashramam
naimisham cheti me sthananyasou mukthi pradhanivai
yetusvashtaksharaikaka: varnamurthir vasamyaham
tishtami krishna kshetre punya sapthakogatha
ashtaksharasya mantrasya sarvakshara maya sada
(I took the English transliteration from the blog you linked)
Here, Sri Ramanuja mentions the 8 places as :
- Srirangam (Srirangam, Tamil Nadu)
- Venkatadrischa (Tirupathi, Andhra Pradesh)
- Srimushnam (Srimushnam, Tamil Nadu)
- Salagramam (Muktinath, Nepal)
- Pushkaramcha (Pushkar, Rajasthan)
- naranarayanashramam (Badrinath, Uttarakhand)
- Naimisham (Nimsar, Uttar Pradesh)
For the location of the 4th place, i.e, totaparvatam, there is a small discrepancy between the two temples, 1. Vanamamalai Perumal Totadrinathan temple in Nanguneri and 2. Lakshmi Narasimha Temple in Mangalagiri.
The Vanamamalai temple is better documented to be the location of the 4th Swayamvyakta temple. (For example in The Lord of Vengadam - S R Ramanujan). From the Wikipedia page:
the temple is glorified in the Divya Prabandha, the early medieval Tamil canon of the Azhwar saints from the 6th–9th centuries AD. It is one of the 108 Divyadesam shrines dedicated to Vishnu.
The reason why Managalagiri temple is also attributed to be one of the eight sacred places is due to its etymology. From Vijaywada MC's website:
The Mangalagiri town has derived from its original name “TOTADRI”. The Mangalagiri denotes that Mangala is Subham and GIRI means a Hill
The official website of the temple also mentions a similar story:
Thotadri is the present Mangalagiri. Lakshmi Devi has done tapas on this hill. That's why it got this name (The auspicious hill).
In conclusion, we really can't confirm as to which is the correct one. This is similar to the location of Vaidyanath Jyotirlinga, which also doesn't have one confirmed location.