First of all: Ananta Dasji sometimes surprises with his translations. In connection to the famous verse from Bhakti-rasamrita-sindhu 1.2.74: guru-pAdAzrayas tasmAt kRSNa-dIkSAdi-zikSANam vizrambhena guroH sevA, he says: “Guru-pAdAzraya is complete in three steps: (1) taking shelter of the guru, (2) being initiated in the Krishna mantra and taking instructions from him about bhAgavata-dharma, and (3) serving him with trust.” I had never seen this interpretation before.
As far as visrambha or vizrambha ("trust") is concerned, I have been trying to find a good single word English equivalent, particularly in relation to the guru, but haven't been able to do so. The Apte dictionary gives: "Trust, confidence, familiar confidence, perfect intimacy or familiarity." His example, taken from Uttara-rama-charita (1.49), is visrambhAd urasi nipatya labdha-nidrAM "In perfect familiarity, she fell asleep on his chest."
Sridhara defines as "praNaya" (BhP 3.20.33), which begs the question exactly what did he mean by pranaya? In Ujjvala-nilamani, as we will see below, however, the word visrambha is an important aspect of the sthAyi-bhAva of pranaya, which follows mAna. Indeed, in the Bhakti-rasamrita-sindhu, the word is used as the principal element in the definition of sakhya, the friendly mood.
Vishwanath uses the word that way in his commentary to BhP. 3.12.29, but specifying that it is the kind of intimate friendship that results out of a familial relationship, which is an interesting addition. vizrambhAt janya-janaka-sambandhottha-sakhyAt.
Visrambha is given as one of the essential aspects of the sakhA. "Krishna's friends (vayasya) dress like him and are equal to him in beauty. They act completely free from his control (samyag-ayantritAH), as they are filled with a sense of familiar confidence in him." (BRS 3.3.8) The point about freedom from control is here specifically used to demarcate Krishna's friends from his servants.
The word visrambha is actually defined in BRS 3.3.106: vizrambho gADha-vizvAsa-vizeSo yantraNojjhitaH--"A particular kind of deep faith [in the friend] that is free from control." Here Jiva says: vimukta-sambhramA gADha-vizvAsa-vizeSo’tra parasparaM sarvathA svAbheda-pratItiH | ata eva yantraNojjhitaz ca, "Visrambha means freedom from fear or excessive reverence (sambhrama). It is a particular kind of deep faith in the other, meaning that friends have a sense of being in no way different from each other."
In the Ujjvala-nilamani, the word visrambha is associated with pranaya, which is the sthayi bhava that follows mana, a kind of intimate state of renewed trust that follows arguments, apologies and forgiveness. Pranaya has two aspects: sakhya and maitri.
mAno dadhAno vizrambhaM
praNayaH procyate budhaiH
praNayaH procyate budhaiH
"When mAna gives way to trust, then that state is called pranaya by the learned." (14.108)
Here again Jiva says: visrambhaM priya-janena saha svayAbheda-mananam "Visrambha means thinking oneself to be one with the beloved." VCT: visrambho vizvAsaH sambhrama-rAhityaM, tac ca sva-prANa-mano-buddhi-deha-paricchadAdibhiH kAnta-prANa-mano-buddhy-Ader aikya-bhAvana-janyaM tatra saty api roSAdikaM tu rasa-svAbhAvyAd eva nAnupapannaM jJeyam | "Visrambha is faith without fear, awe or reverence. This feeling results from a sense that one's own life, mind, intelligence, body, clothes and everything else are one with those of the beloved. In this state, due to the quality of the mood, anger cannot arise.
svarUpaM praNayasyAsya
vizrambhaH kathito budhaiH |
vizrambho’pi dvidhA maitraM
sakhyaM ceti nigadyate ||
vizrambhaH kathito budhaiH |
vizrambho’pi dvidhA maitraM
sakhyaM ceti nigadyate ||
The wise say that the essence of this pranaya is confidential trust. Such trust is of two kinds: maitram and sakhyam. (14.110)
bhAvajJaiH procyate maitraM
vizrambho vinayAnvitaH ||
visrambhaH sAdhvasonmuktaH
sakhyaM sva-vazatA-mayaH ||
vizrambho vinayAnvitaH ||
visrambhaH sAdhvasonmuktaH
sakhyaM sva-vazatA-mayaH ||
A familiar trust that is accompanied by humility is called maitram. While the same kind of familiar trust that is completely free from any kind of caution and brings the beloved under one's control is called sakhya. (14.111, 114)
So, on the whole, when the term is used in the phrase "visrambheNa guroH sevA," what was Rupa Goswami getting at? It would seem to me to be not the kind of belief or faith that is usually spoken of in relation to the spiritual master, but a sense of trust, of oneness, in the sense that the disciple believes the spiritual master has his best interests in mind. As Ananta Das says, "Another valuable result of this is that through such sincere service to a saint, the guru principle itself automatically melts with compassion towards the disciple." In other words, one serves the guru principle in the confidence that the channel between disciple and guru tattva flows through this human being that one calls "guru."
However, to expand on this point, one needs to look at the words zraddhA and vizvAsa in greater detail, which expands the discussion considerably. For instance, where sakhyam is considered one of the angas of sadhana bhakti (BRS 1.2.188), it is subdivided into two, vizvAsa and mitra-vRtti. BRS 1.2.191 distinguishes zraddhA from vizvAsa, as a "before and after" sort of thing. Sraddha is what gives one the qualification to engage in bhakti, vizvAsa is a bhakti-anga.