You asked a few related questions.
Question (1): Then, what is the difference between collateral and pledge when buying a home? Are they equivalent notions in this case? Is there a different financing situation where two are clearly distinct?
"Pledge" and "collateral" in the manner that you're using them do essentially mean the same thing. You're pledging the home as collateral for the loan. This basically means you're giving the lender a lien on your title to the property, which means they have rights to it's value until the loan is repaid and the lien is released.
This is true regardless of if the loan is a so-called first position mortgage (meaning, it is the first lien on the title and that lender gets "first dibs" on the value), or if it's a junior position home equity loan (as in the example of borrowing a $3k amount against a property you already own and may already have financed elsewhere).
Question (2): The resale value of the depreciated home or current home? When a loan officer assesses my financing needs for buying a home, does he or she look at the resale value of my current home price if appraised at the time of transaction or far in future, and compute the present value of the resale value of the depreciated home's worth?
When a loan officer is considering you for a loan, they'll typically require an independent third party to appraise the property for a first mortgage. This way, they have established the "value" of the collateral at that point in time. Typically there is zero consideration paid to trying to determine the value "far in the future."
Practices for junior liens (your $3k example) vary in terms of how, or if, the home's value is appraised. Typically, the lender will have a loan to value threshold based on a passive assessment (i.e. the current tax assessment) and if that threshold is exceeded, they require a new appraisal. So, for example, if you own a $300k home outright with no financing, and you ask for a $3k loan, they'll likely give it to you. But if you owe $250k on the home and are asking for $30k, they will likely need a new appraisal.
Just to be clear - you seem stuck on "depreciated value" or "future value" or other terms about the value. From a lender's perspective, there's really only one value that matters, the assessed value. Lenders don't care that the home used to be worth $500k and has (somehow) depreciated to being worth $400k, or that it's worth $300k now but someone thinks it'll be worth $400k in 10 years. They care what it's worth now with the exception that sometimes their risk is low enough that they'll use a stale value for "now."
Question (3): Is it possible for a typical home to be divided into a pledgeable portion and non-pledgeable portion? I can only think of a situation where a home would NOT be pledgeable in the future is when something like flood or tornado destroys the home completely, so the physical estate is ruined to an extent it is no longer pledgeable? Can anybody share a different scenario if has any?
It's not typical for a home to be "divided" in exactly the way you're describing. If the home is destroyed, there's no longer any value (except that which is recovered via insurance).
The only ways in which a home's value is typically "divided" are when a lender requires a certain loan to value ratio - so if your home is appraised at $300k, they may say the most you can borrow is $295k. In a sense, that "divides" the property into a pledgeable $295k value and a non-pledgeable $5k value. But that seems like a slightly different meaning than the "division" you're asking about.