The official term for the white light on the front of the mast, used only when under motor power (whether or not the sails are up) is "Masthead Light", but it's more commonly called a 'steaming light' in casual conversation. I always write it as "Masthead (steaming) Light", to cover the official and the common-usage terms. To confuse things, some folks refer to the all-round white light on top, used as an anchor light if it's there, as a masthead light, but that's not correct - the term has an official use, and that ain't it. Being on top of the mast doesn't make it a masthead light - it's a term referring to function rather than location, at least these days, and in official rules and regs. Historically, the 'masthead' isn't at the top of the mast anyway.**
The Masthead (steaming) Light is always white, 2-mile (or more, for longer vessels) intensity within a certain vertical azimuth, faces forward, and sweeps an arc from dead-ahead to 22-1/2 degrees abaft the beam on either side. Same arc of coverage (horizontal azimuth) as the Sidelights*. The Masthead (steaming) Light must be one meter or more higher than the sidelights for a small boat like ours, but can be higher. My keel boat had the Masthead (steaming) Light at the top of the mast (see below).
The Masthead (steaming) Light is used only when under power, even if the sails are up (you're a power boat when motor sailing), but if under sail power alone (engine off), you turn it off. That's why it has its own switch, unlike power boats, which are always power boats, and always burn all nav lights when under way. On a sailboat, you only burn the side lights and stern light when sailing without the engine running.
An anchor light is merely an "All-round" (official term) white light. It should have an unbroken arc of 360 degrees,
but it can be broken by up to 6 degrees (that's in the COLREGS), which allows for an anchor light hanging in the foretriangle but obstructed by the mast, for instance. There's also other language giving relief on anchor light location based on the practicality of various locations.
The Anchor Light actually
can be two lights within a certain distance of each other (there are specs in the COLREGS) such that from a distance they appear as a single light. My last boat actually had only a 2-element white light on top of the mast - Masthead (steaming) Light using the front bulb, and Anchor using both. The stern light was on the hull.
https://www.westmarine.com/buy/perko--f ... cordNum=35 I see these a lot on the Erie Canal rental barges/boats, which are long and low, as they have to put the Masthead (steaming) Light up higher than the Sidelights anyway, so the second element just makes it an anchor light.
*Proper classification for the bi-color bow light is "Sidelight(s)". Small boats are excepted from having separate Sidelights (though they can), and they can be combined into a single bi-color light on the bow, on the centerline. But for anything longer than 12 meters (I think) has to have separate Sidelights, and there are even rules about 'screens' to prevent bleed-over beyond dead-ahead, so you don't see both red and green anywhere but directly dead-ahead.
There are even COLREGS specs about the vertical azimuth for Sidelights and Stern Lights, which are different between power boats and sailboats (sailboat lights cover a greater angle, since they heel while under way), and how fast the intensity drops off with heel angle.
And that doesn't get into single tri-color lights for sailboats under sail, or the additional red-over-green lights at the top of the mast of ocean sailors ("Red over green: sailing machine!), which are in addition to the standard Sidelight and Stern Light requirements, as another vessel wouldn't know from the two lights at the top of the mast what your orientation is unless you also burn the Sidelights and stern Light (which the other vessel will see once closer, especially in a seaway, so the two at the top are an early warning system that you're a sailboat).
Had enough? Or should I keep going?
It's complicated.
Edit: I should clarify that you
don't burn the navigation lights and the anchor light at the same time. Nav lights are as much for showing orientation as for visibility, so other vessels know what action they should take (stand-on vs give-way vessel), and burning nav lights when anchored could confuse someone, especially if they're the stand-on vessel and are expecting you to give-way, when in reality, you're anchored and three sheets to the wind and just watching as they almost run you down because you didn't 'give way'. Or something like that. If all they see is a single white light (and no red or green sidelight), it's because you're either moving away, or anchored, and in either case, the other boat is the burdened vessel (I think).
** Wikipedia wrote:Masthead
A small platform partway up the mast, just above the height of the mast's main yard. A lookout is stationed here, and men who are working on the main yard will embark from here. See also crow's nest.
Crow's nest
Specifically a masthead constructed with sides and sometimes a roof to shelter the lookouts from the weather, generally by whaling vessels, this has become a generic term for what is properly called masthead. See masthead.
That's almost certainly where the term "masthead light" came from - a fuel lantern hung from an accessible location on the mast, the crow's nest or 'masthead', which eventually became what it is today - an electric light partway up the mast to indicate a vessel under power (in concert with red/green sidelights). Big boats (over 50 meters) use two of them, with the second one behind the first, and higher.
(I dunno) but that's a light for sailing at night? ( 
