If I’m using my 16 mm camera, I’m filming. If I’m shooting with a mini-DV camera, I’m taping. What am I doing if I’m recording video onto the flash drive of my DSLR? Just ‘recording’? ‘Videoing’?
I’m in TV/documentary. We mostly say “shooting,” sometimes “filming” even when not using film. If referring to an active process we say “rolling,” as in “were you rolling on that?” Sometimes “speeding,” most often at the beginning of a shoot or scene: “Are you speeding?” “Sound speeds.” “Camera speeds.”
ETA, you only hear “taping” or “videotaping” rarely these days, unless you’re talking about actual videotape.
I generally just say “recording.” However, I occasionally hear people call it “taping,” even when there’s no tape involved.
Here is someone who calls it “videoing.”
So is sound recording “audioing”? 
That still seems to get “tape” as well. “Cut” does seem to have long since gone.
For sound we’d just say “recording”, whether in studio or in the field, or “sound recording” if a distinction needed to be made.
“Filming” or “taping” seems to be in common usage to describe what nowadays is neither, but videorecording onto a memory medium like a flash drive or a magnetic hard drive. No one who asks me to video record ever uses the correct terminology. I’m torn between correcting them and letting it go. Maybe the dictionary definition of “taping” will change soon.
It usually clanks off my ear when I hear someone say “filming” when they’re not using film. And when someone says he’s “filming” with his iPhone I die a little inside. (Maybe because I go back far enough to have shot regularly on 35mm) “Recording” or “shooting” is best, in my opinion. When we shoot on XDCam (re-recordable DVD) we’ll jokingly say we’re “discing.”
Oh, and for the love of Christ turn your phone sideways when recording video.
Flashing?
Do you still dial a phone? Hang up? Use a Shift key? Send a CC?
Don’t get me wrong, I know there are countless anachronistic terms that have made the leap to refer to modern technology. I’m sure I use them everyday. Maybe it’s because I work in the industry that I’m a bit of a pedant about it. Or maybe it’s because alternatives to dialing or shifting aren’t as apparent.
For what it’s worth, “shift” doesn’t have to refer only to the mechanical shift of a typewriter. You’re still shifting between functions of the same key. (Did I mention I can be pedantic about word usage?)
Only while wearing a long coat …
In a non-professional, casual environment I would always say ‘recording’ when using any kind of solid state (or hard drive storage) digital camera. Even though there’s no longer any ‘tape’ involved I might still also call it ‘taping’ because unless your digital camera has the capability to record at 24fps (and it’s set to do that) the footage is going to look like videotape, the main aesthetic difference between the two being the frame rate (film is 24fps, videotape is 30fps). I would never say ‘videoing’ because it’s just a cumbersome word (just now I was surprised that my spell-check didn’t flag it, didn’t think it was even a real word!)
You would never refer to using any kind of film camera as ‘recording’ because recording is an intrinsically electronic process. Even after they were no longer cranked by hand movie cameras often still didn’t contain electric motors, they were driven by hand-wound springs. My father had a 16mm home movie camera in the 60s and it didn’t use batteries.
The closest example in which you might use the term ‘recording’ with a film camera would be in an ethereal sense, if you were filming an important event you could say you were ‘recording it for posterity’…
I love the term “voice dial.”
I use all these anachronisms, and it still bothers me to refer to “taping” in reference to using the DVR. Sometimes I use “DVR” as a verb. Usually I say “record.”
I would never use ‘recording’ to indicate that I’m filming.
I also have some wind-up Bolex 16 mm cameras, but my ‘real’ cameras are Aatons. They’re silent, so they can be used when recording sound.
When my wife or I store content on our TiVo’s hard drive, we are “recording.” When my elderly parents store content on their TiVo’s hard drive, they are “taping.”
Keying?
I use the term “shooting”, but avoid using it when I’m recording one of my School of Rock shows. You don’t want to tell someone that you’ll be shooting their children.
I use “cut” for editing, primarily because my style is almost always doing straight cut transitions from one recorded camera to another.