Mortgage lenders are interested in four main factors when considering a loan:
- What is the value of the home? The home will be collateral on the loan. Lenders don't want to lend so much that they will be upside down (or even anywhere near upside down, ideally) if you default. Lenders will assess the value of the house to determine this, and compare it to the total amount you are borrowing.
- What is the likelihood that you will miss payments, be late on payments, or generally be delinquent on paying the loan? Lenders want to be paid consistently on time. In other words, how risky are you? They will use your credit score to determine this, but they will also look at the details of your actual credit report, and specifically they will look at payment history for other home loans you've taken out in the past.
- What is your cash flow? Can you afford to make payments each month? Note: this is a different issue than the prior point. Many people who have plenty of cash flow will still default on loans (it's hard to understand, but it happens). So, they will want to know what your income is, and what your other monthly debt expenses are. They will get information about your debt from your credit report, and they will ask you for proof of income (your credit report does not include income information). These two numbers will form your DTI, your debt to income score. They will add the theoretical payment for your new home and see if that puts you over their target DTI ratio.
- What is your current cash (or liquid assets) on hand? Lenders want to be sure you will have sufficient cash to close the loan. Closing a loan includes a down payment in most cases, but typically also many other expenses - you may have to pre-pay property taxes, insurance, and so on - plus you may have to pay closing expenses to the lender, or a lawyer, or pay for filing a title, etc. Lenders want proof that you will have cash on hand to do this. Generally, they will ask for bank statements or proof of balance in your accounts to determine this.
I'm speculating, but I would assume that last point was the reason for the advice you got. If you are trying to obtain a loan, but you don't have a lot of cash on hand, you may be turned down. If you're paying extra on your current mortgage each month, you could theoretically start putting that extra cash in a savings account instead, to show that you have cash on hand.
On the other hand, the fact that you're paying down your current loan means that the portion of your current home's sale price that you receive as cash is increasing, so in the end it may be a wash - you end up with the same amount of cash on hand either way. However, in this viewpoint, it's important to consider that the sale price of your current home is a factor. If you're pre-paying all that cash into the current loan, and then your current home sells for much less than you expected, you won't have that cash on hand.
How a given lender interprets these scenarios will vary, and - of course - it will also be highly dependent on the details around the sale of your current home. In many areas, it's common to obtain a new mortgage based on the fact that your current home has a certain assessed value and will leave you with X amount of cash (after your current loan is paid off). Many lenders have a standard formula they apply to their "cash on hand" determination when including cash obtained from the sale of a prior home - they may derate it a few percent or otherwise account for the potential variability.
All that said, it's very difficult to address specific advice to your specific condition, since we don't have all your details and none of us are your lender. In the end, the best approach is to talk to your lender about their requirements. Many lenders will be happy to walk through these concerns with you, and offer you advice on how certain actions you can take will influence their decision process.
Also - regarding your second quoted block above - you are right, it's generally bad advice to try to take out any loan immediately before trying to obtain a mortgage - especially a home equity loan. Lenders will be nervous that you're trying to play games, or hide the fact that there's something else going on (i.e. are you taking that loan out because your current house is un-sellable and needs some sort of major work?) In addition, if your current house is already listed for sale, many lenders will be very hesitant to give you a home equity loan against it (since you're pretty much advertising the fact that you're about to sell and pay the loan back, which would mean the bank takes a loss for originating the loan and has no opportunity to make money on it).
Further, if you use the home equity loan simply to inflate your cash on hand, lenders will be suspicious. Mortgage lenders like to see that you "naturally" have a lot of cash on hand because of good financial habits. If you walk in with a bank statement showing that your savings account suddenly had a very large deposit, they will want to know exactly where that money came from. Telling them that it came from a loan will be met with concern.
Editing to add: Referencing your financial situation as described in your other question, it's important to note that cash on hand and cash flow are different factors. Taking out a loan in an attempt to say "look, I have a lot of cash on hand! I can definitely afford this new loan!" will not work. It will not change the fact that your projected DTI once you are in your new home won't meet the bank's requirements. And it's useless to try to modify your DTI by playing shell games with loans against your current home, because lenders will take any loan backed by your current home (which you're about to sell) out of the DTI equation. When a lender is trying to decide if you can afford a monthly payment, they do not consider cash on hand OR loans against the current home as part of the equation.