Belarus-Russia border closed to foreigners

The main point of the news you're talking about is that foreigners should use border checkes during traveling between two countries, and that there are no such borders checks while traveling by car or train, because of "union" of the two states, so, even if you have sufficient documents to get back into Russia, there could be no person able to check them at the border. So, you could try to get back to Russia by plane, as there is border checks in the airports (still may be not an option, you should contact the officials for this).

Update:
Some decisions of Russian aren't understandable. More on that: there is a train from Ukraine to Russia via Belarus, and foreigners can use that because there will be a border check at the end. But if you're traveling by car, no one can guarantee you will pass border because Russia set up different rules for foreigners and Belarus residents. No-visa agreement was setup only for Belarus residents, as it appears right now, not for everybody.

Belarus-Russia border crossing should be fully legal as of 2020-07-01

As of 2020-07-01, Belarus and Russia reportedly recognise each others visas:

Foreign Ministers Vladimir Makei and Sergey Lavrov signed the agreement on mutual recognition of visas between Belarus and Russia in Minsk on Friday, 19 June.

Belarusian Foreign Minister Vladimir Makei explained that the document provides for full mutual recognition of national visas and other documents related to the entry and stay of foreign nationals.

“Now foreign citizens have the opportunity to visit both countries with the national visa of one of them. Thus, the agreement removes the problems of third-country nationals crossing the Belarusian-Russian land border,” said Makei.

Under the agreement, foreign citizens with a Belarusian visa will be able to freely enter Russia and, vice versa, holders of Russian visas will be allowed to enter Belarus. Foreign visitors will reportedly be able to travel by air, rail and car.

Source: Belarus And Russia Sign Agreement On Mutual Visa Recognition, Belarus Feed, accessed 2020-07-05. Also reported by the Belarusian Telegraph Agency, TASS, and intourist.

Warning as of 2020-07-07: another source in Russian, dated 2020-07-01, states it will take several more months until it actually takes force. These sources contradict each other. Either it's already in force, or it will be soon (and as of July 2020, the 2019-2021 Coronavirus Pandemic means few travellers will make this trip anyway).

^{2019-2021 Coronavirus pandemic note: Russia and Belarus may still have different COVID-19 entry restrictions, but this is independent of the new visa regime. At the time of writing (2020-07-05), most foreigners are banned from entering Russia but allowed to enter Belarus, and the Russian-Belarusian border may still be closed to everybody. Now is not a good time to travel as a tourist to either country, but the new visa regime is good news for international travellers once the situation normalises.}

there is no passport control at the airport in Moscow if you go to Minsk ! You need a Russian visa if going through Moscow/ St. Petersburg to Minsk. I have just talked to the Russian embassy in Berlin regarding this... they do not know anything. A regulation like that would result in "no direct travel between Russia and Belarus, because there is no control in trains neither

Both VMAtm answer and Skott K assumptions are wrong here.

So, you could try to get back to Russia by plane, as there is border check in the airports.

Yes, there is border check in the airport but you won't be allowed to enter Belarus anyway. Russian visa gives no allowance to do that and Belarusian Consular Office in airport don't make visas to foreigners who arrive from Russian flights.
So you won't be able to obtain visa even in airport, you should get it prior to your arrival!

All possible grounds when you are excused from obtaining transit visas are listed in paragraph four of the Resolution of Council of Ministers of Belarus about transit, and RU-BY flights (and vice versa) are not among them.

If you are not eligible to any of this grounds you need Belarussian visa (either transit or entry), no matter of way you arrive Belarus (ground or air).

UPDATE: what concerns Russian or non-Russian planes (as suggested by pnuts), Russian Embassy in Belarus gives zero clarification about this:

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Passengers who use flights from/to Belarus through Russia should pay attention to the requirement of transit Russian visa disregard of duration of stay in Russian airport, whether they fly by federal (domestic) flight from Russia to Belarus or vice versa.

There is NO mention about carrier of flight or whatever.