Ctrl+L stopped clearing screen in PuTTY sessions

If you go into the Putty Configuration screen, under the "Terminals" category, in the "Features" tab, there is a checkbox for "Disable switching to alternate terminal screen". Make sure to check it, and you'll get exactly the behavior you're looking for.

From the Putty Documentation:

4.6.4 Disabling switching to the alternate screen

Many terminals, including PuTTY, support an ‘alternate screen’. This is the same size as the ordinary terminal screen, but separate. Typically a screen-based program such as a text editor might switch the terminal to the alternate screen before starting up. Then at the end of the run, it switches back to the primary screen, and you see the screen contents just as they were before starting the editor.

Some people prefer this not to happen. If you want your editor to run in the same screen as the rest of your terminal activity, you can disable the alternate screen feature completely.

Source

The terminal emulator code used by MobaXTerm (which seems like it might have been taken from PuTTY) doesn't support that key combination and doesn't send any keycode. There does not seem to be any configuration available in MobaXTerm for its terminal behavior (not even the ability to assign key shortcuts for macros), so you can't really fix this except by contacting the MobaXTerm developers and asking them to add it.

(The key code for Ctrl+Del follows the same "Xterm modifiers" format as Ctrl+PgUp/PgDn, being ESC[3;5~ while the base Del keycode is ESC[3~.)

I have tried all possible terminal type values like xterm, xterm-256color, putty and more, but no luck.

These options only set TERM values that are used to inform programs what to expect from the terminal. They don't alter the way the terminal itself behaves. If it doesn't send anything for a particular key, no amount of changing TERM can make it do so.