The trains you indicated (e.g., ICE 1612) are express trains (ICE/IC/EC) of the Bahn. They are not covered by Länder-Tickets (which are limited to regional trains), but they are covered by standard or by SparPreis tickets.
In addition to standard (full fare) tickets, you have a choice of:
SparPreis (or Savings Fare) tickets. These are advance purchase, online tickets that start at 29€ from anywhere in Germany to anywhere else in Germany. The ticket must include at least one leg on a train of the Bahn (ICE/IC/EC) and those trains are specific (date and time) to the ticket (i.e., the ticket is not valid for any other train on that route). Tickets go on sale at the lowest price 92 days before the date of travel. The tickets are "tiered"; when the lowest priced tier sells out, the price goes up. On some popular routes, the lowest priced tickets sell out in hours, if not minutes.
Länder-Tickets, or regional tickets (such as the Bayern-Ticket in Bavaria), are all-day passes for a Land (state) in Germany. They are valid for unlimited travel on regional trains in that state¹ from midnight weekends or 9 AM workdays until 3 AM the following day. Länder-Tickets never sell out; they are always available for the set price from automats in the train stations or for 2€ more from a ticket counter.
¹ Länder tickets often include travel to a "border" station in another Land or country, or loops of track just outside the Land. In some cases multiple Länder (such as Hamburg with a Niedersachsen ticket or Sachsen, Sachsen-Anhalt, and Thüringen as one ticket) are included in one ticket.