PEC cleaning pads, Eclipse Lens Cleaning Fluid, Hydrogen Peroxide, Acetone based fingernail polish remover.
Naptha, preferred, is the old Naptha Ronson lighter fluid, but the VM&P stuff the hardware shops carry will work too.
PEC Pads plus Eclipse will give you a streak free result, but you should take some youtube instructions on how to use them, as well as how no to, and I recommend Mikeno62 videos, in which he offers some very good repair instruction.
Add some Harbor Freight black nitrile gloves, more tools and consumables depending on how far into a lens you wish to delve.
A brief rundown is, the Eclipse fluid is touted as the one lens fluid recommended by all lens (glass?) makers and PEC Pads are the perfect wipe, do no use other cleaning cloths, papers.
In the example photos, your technique shows very clearly and the major issue, besides wearing clean, nylon, silicone, or nitrile gloves, both hands, is how you end your wipe, simply lifting off the lens, when what you need to do is use the corner of the PEC cloth to PULL the pad away, straight off the edge of the lens, taking the last of the damp fluid and freed dirt/grime with it.
Practice cleaning with a PEC cloth, on say a plastic jar lid, and as you reach the outer side of your circling cleaning action,pull your finger away from the side or bottom in a 90° line with a lifting action to help carry the dirt away, in a slight downward action (if the lens is free of the barrel/ring, or upward over the metal lip.
If your Eclipse fluid does no clear away fungus, try the Hydrogen Peroxide; I use it by putting a PEC Pad in the bottom of a plastic jar lid, and adding a shallow depth of H2O2, so the lens surface is in complete contact with the one side of the lens, then do the other side as needed, letting the lens soak for a few minutes, as needed.
Allow the lens, both sides, to dry, and then use the PEC Pads and Eclipse to clean first one surface, then the other, pulling off the glass as I (really Mikeno62) suggest.
In using the PEC Pads, with Eclipse, only a wet an unused corner of the pad each time, and the edges between the corners, and center.
I have been flipping the cloth over, say the corner used to remove the first layer of grime, reweting the reverse, corner and gentle wiping away the last of the dirt, in the doing.
So far so good, but I want to get the best use out of these pads, and the gloves keep oils, debris, away from the lens(s).
Naptha can be used to clean dirt away, but be gentle, and do no forget how to draw away your fingers/cloth.
Acetone, polish remover, IN TINY AMOUNTS can be used, on threads (only) to help remove old lube, or glue, to free the lens when it's stuck after all other screws, rings, etc, are moved out of the way.
I am concerned Acetone cleaner will strip away coatings, so keep it off your lens's surface glass, and especially the black paint on most lens sides.
I don't think this should be allowed to clean the lens and should be wiped away from the black painted Len's edges, gently to avoid stripping the paint away.
This is no such a brief description, but these things work.
By-the-way, I have used Eclipse on B&W, Hoya, Hasselblad, Tiffen and Helios coloured filters with no sign of marking or surface damage to the colourent or rings, but do keep the possibility in mind when cleaning other filters.
Watch Miken62's videos on cleaning lenses to see the correct way to approach lens cleaning, and never worry about doing a bad job of it again.
IMO.