The "by January 15th payment" is indeed on the worksheet for Form 1040 line 30. In the IRS instructions, the worksheet on page 59 has it on line 8 (out of 21) and refers to it as "EIP2". You might also see reference to Form 1444-B, which is the notice of payment that you get telling you how much your EIP2 amount was.
If your tax software update is not yet making reference to two different EIPs, I would wager that you should check for another update and not file until you get a worksheet or data input screen with 2 EIPs on it.
It's not on the H&R Block tax software that way. The mini worksheet for line 30 is for $2,400 only.
Step e; Enter $1,200 ($2,400 if filing jointly)---This step establishes how much rebate you may be eligible for. I left out the steps for kids and AGI calcs.
Step m; did you get an economic stimulus payment in 2020? If yes, enter in line m.----This step establishes the amount you received in stimulus based on the amount you are eligible for as calculated above.
Step n; Recovery rebate credit.---This step establishes the difference between what you got and $2,400. If I put down I got nothing, then this box is populated with $2,400.
For the above steps, I played and entered $2,400 and step n/line n was zero. If I changed that to $1,200, then line n was $1,200. The combination of m and n, what I received and what I would get for credit always added to $2,400.
So the 2nd stimulus, the $600 per person, is not on the 2020 taxes according to H&R Block software.
I understand that this should likely get updated at some point before the final allows me to file my taxes. BUT, since I already went through these steps, an update wouldn't trigger me to have even checked. I will now because of what was posted here, but most likely if I were not a member of this site, I would have only done the first stimulus and not the 2nd.
BTW, if I don't claim I got the $2,400, my taxes change from owing $1,964 to a refund of $436. That means the credit gets applied to owed taxes. They do not keep the stimulus separate as a payment from your taxes owed.
One last thing; H&R Block software says the next update will be January 20th.