There were a few iterations:
1914
This note is receivable by all national and member banks and Federal Reserve Banks and for all taxes, customs and other public dues. It is redeemable in gold on demand at the Treasury Department of the United States in the city of Washington, District of Columbia or in gold or lawful money at any Federal Reserve Bank.
Redeemable in gold on demand at the United States Treasury, or in gold or lawful money at any Federal Reserve Bank.
The version you noted, which referred to "lawful money" (whatever that is) but not to gold, appeared in 1934. That phrase was replaced in 1963 with the words we see today, "This note is legal tender for all debts, public and private."
Article 1, Section 10. (US Constitution) No state shall enter into any treaty, alliance, or confederation; grant letters of marque and reprisal; coin money; emit bills of credit; make anything but gold and silver coin a tender in payment of debts;...
I reckon that would sum up "lawful money".
While Federal Reserve Notes early on were redeemable in gold there were also Gold Certificates, which stated on the note that "This certifies that there have been deposited in the Treasury of the United States of America (denomination) in gold coin payable to the bearer on demand." It also said "This certificate is a legal tender in the amount thereof in the payment of all debts and dues public and private"
After the 1928 series and after the confiscation of 1933, there was another series of Gold Certificates in 1934 used in transactions between the US Treasury and the Federal Reserve Bank, which said "This is to certify that there is on deposit in the treasury of The United States of America (amount) in gold payable to the bearer on demand as authorized by law. 1934 $100 Gold Certificate
So, yes, it is complicated.
The 1914 Federal Reserve Note merely said that the note was "Authorized by the Federal Reserve Act of December 23, 1913" image of $10 FRN and elsewhere on the note said that the "United States of America will pay to the bearer on demand (amount of denomination)". This was, in essence a promissory note: an IOU.
Your post of the Federal Reserve note of 1928A is correct, and the simultaneously issued United States Note had the message: "This note is legal tender at its face value for all debts public and private except duties on imports and interest on the public debt". 1928 United States Note $1, but it also said the United States of America would pay to the bearer on demand one dollar.
That all of these notes, in one form or another mentioned redemption in "lawful money", Gold, or silver, or noted payment would be made upon demand of the face value amount implies that they are mere promissory notes to pay in the Constitutionally lawful coin of gold or silver.
At least that's what it looks like from the cheap seats. (YMMV)
