Well back in my day, '79-'90 in the US ARMY. We could fire mortars "direct" by using "Open sights" ... i.e. you could see your target. So in effect you ARE firing the mortar "direct" since you can see the target. Regardless of it's a high angle trajectory. I've done it, but only rarely ... Of course the higher angle with the right elevation and charges will certainly increase its range. That is one capabilities of modern artillery and mortars.
As with the Howitzer, it generally fires at a high angle to go over terrain, etc. And generally does not see it's target. As with mortars they generally don't see their targets either. They are called in by an FO of some type. As a Rifle Plt Leader and later Mech Co. Cdr, etc., I was capable of call-in mortars, FA, CAS, Gunships and even Naval Gun Fire.
The same goes with all Field Artillery, whether howitzer or cannon, their firing mode generally is in indirect high angle trajectories. But it was not unheard of in WWII, the Korean War and Vietnam. For all forms of Indirect Fire weapons would/could fire directly at their target. I.e. they can see the target and may have to fire "open sights". The trajectory of the rounds is not a factor. I.e. mortars will fire using open sights, and the rounds trajectories will still be at high angle.
The US WWII 60mm Mortar in many cases would fire direct. As with it's relatively short range they may see the attacking targets. It was kind of like a Grenade Launcher, e.g. M79 and M203 ... But regardless if need be the WWII 60mm mortar could fire indirectly when called in by an FO, etc. And did ...
Even the old 105mm howitzer had a HEAT round just in case it had to fire directly at enemy armored vehicles. E.g. it happened during the Korean War.
In Vietnam the 105mm howitzer had a Flechette round which turned the weapon into a very big shot gun. Usually firing directly into massed Infantry attacks by VC or NVA.
Interestingly even the quad .50/M2 .50 MG could be fired indirectly as well. As it has a very long range even with a flat trajectory, IIRC e.g. 6800m ... It again was used in Vietnam, e.g. at Khe Sahn. But again it usually could see it target. But again ... not always ...
Being an indirect fire weapon, e.g. mortar, howitzer, does not relate to the trajectory of the rounds ... but whether they can see the target are or not. If they can see the target it is direct fire again regardless of the rounds' trajectory ... Flat or high angle ...