Confused about Deutsche Bahn system

do I need to buy new tickets or just proceed to ride the alternative
train that they have?

In your own interest I recommend you shall contact Deutsche Bahn directly for a legally reliable answer.
Any opinion from a forum is just an opinion and not binding for Deutsche Bahn.

In your own interest I recommend you shall contact Deutsche Bahn directly for a legally reliable answer.

Don't do that, you'll be on the phone for hours with no results. First of all, the email you have received is "legally reliable". Secondly, given the current construction situation on the DB network, timetable changes are extremely frequent, and the DB's rules on this are crystal clear (https://www.bahn.de/service/fahrplaene/benachrichtigungen-zur-reise/verspaetungsalarm_fahrplan , translated):

"The train binding [of a ticket] is suspended if there are changes to the timetable in the run-up to your trip - for example, due to works on the rail net. If you have bought a ticket with train binding, the train binding is suspended:

• if there are changes to the timetable and the departure time at the starting point
• if you arrive at your destination more than 20 minutes later than originally booked due to the schedule change
• if the train or a stop is cancelled
• if the connection between departure and destination is cancelled

You can then use your ticket for other Deutsche Bahn trains to reach your booked destination."

Usually, notifications like the one you received are marked as " suggestions" for a new connection. This is not binding, you could also take another one as long as you don't make any big detours (e.g. via Munich ;)).

Overview of your passenger rights in English here.

In a similar situation last year, I went to the Travel Office in a railway station, and they changed my tickets free of charge. I did not use the mobile app, but had paper printouts of the tickets.

I am still seeing plenty of direct trains to Cologne Hbf from Berlin Hbf on Aug 4, including a Sprinter at 11:42 that will get you there in 4 hours. What was the original departure time, and did you purchase a seat reservation or buy a 1st class ticket?

I agree that you could easily clear this up when you get to Berlin.

Per their reply as stated by you. It says that you can now use your ticket for other DB trains. There you go.
What is the worst that could happen if your present ticket is challenged?

This is what many people often fail to realise. Train travel is different from air travel.

When you buy a train ticket from Berlin to Köln what you buy is not a seat in a particular vehicle. DB is not in the business of selling seats. They sell transportation. You have bought a transportation service. And a standard ticket from Berlin to Köln allows you to take any train on that route for a whole day. So you are free to leave when you want, on a train of your choosing, and the railways don't even know in advance what train you are going to be on (nor do they care). So in short, your ticket is for travel, not for a train.

Its like mass transit, a subway, but nationwide and with nicer trains.

DB (and other railways) do nowadays also sell discounted tickets, where you commit to taking a particular train, and in exchange get a reduction. They do that because they see that some of their trains are fuller than others, and want to steer people towards the lesser used trains. But as has been pointed out: When the train your ticket is tied to as changed your ticket reverts to the default, which is "valid for any train on the route".

If you have reservations for that train ask for a refund and make a new one. You can do that at the station if you want. Without a reservation you do not have a guarantee to a seat.

Note also that this is a typical example of a problem that mostly is experienced by foreign tourists that book train tickets months before anyone in Germany does. When you will board your train on the 4th most people on that train will have bought their ticket just minutes before departure... Booking months in advance is not something Germans do, and as a result the system is not really set up to deal with the situation that arises when a train that was booked months ago is not no longer running.

Something similar happened to me in like that. The train was rerouted, and I was going to have to make another stop, get off, take a bus, and then pick up another train and so on. I opted to leave earlier and made it with just one transfer. As someone above mentioned, the alternate routes are just suggestions. You can change it as long as you are going in the same direction and to the same destination. And you don’t need to change the ticket. Just keep the original one and be prepared to explain what happened. When the conductor saw my ticket, I started to explain, and he just smiled and said it was no problem. 😊

What is the worst that could happen if your present ticket is challenged?

That will not happen. The conductor can verify that your connection has changed and that the train binding for your ticket has been removed.

But if you want complete peace of mind, make a paper printout of your ticket and get a beautiful old-fashioned stamp "Zugbindung aufgehoben" ("train binding removed") on it at the information desk of the train station.

sla019, I do not share your opinion. This will not help OP because it is more than train binding. The travel distance is longer; therefore it can happen that an additional price is charged. It does not matter if this is wrong or right because in a 1-on-1 situation in a train in a foreign country with foreign language the OP will not do the discussion.

Again my recommendation: Just write an e-mail to Deutsche Bahn or use the contact form. No need to trust a forum opinion or to wait in a phone line. Traveller's life can be simple.

because it is more than train binding. The travel distance is longer; therefore it can happen that an additional price is charged.

I'm sorry, but that's completely wrong; the price does not change. The OP has received legally binding information about this and will not receive any other, even in the case of an email contact, to which, as experience shows, he will only receive an answer after a long time.
To put it bluntly, if you yourself obviously have little experience with the problem being addressed, it might be better not to increase the OP's confusion with useless advice. At least you should have provided an email address.

I've had this happen on a Dresden-Munich train recently. I just kept the original ticket on the app and took the recommended new train and everything was fine when they came to check the tickets. Just like the notice you received "You can now use your ticket for other trains of Deutsche Bahn".

DJ