1/8" or 3/32" lathe bits cost a few bucks a piece.
I've found them to be the easiest to make engraving tools.
Yes you can make a tool out of anything from a piece of car spring to an old file, and they can do great work. .
But you've nothing to prove in doing so,,you are trying to learn the engraving trade.
Don't handicap yourself both in time spent and possible tool quality.
I've made many more than I'd like to think about in the first few years. HSS pre-shaped gravers are OK too but just don't hold up very well on the modern HT'd steels.
Unless you just plain enjoy grinding, shaping, heat treating and such & perhaps are making only a couple of tools,,I'd use the small lathe bits or something close to them. All sorts of different alloys to choose from.
When I showed my prized hand made gravers to a pretty fair engraver by the name of McKenzie when I was just starting ,,his first words were to"Toss these points out. You want to be a professional engraver,,not a professional tool sharpener".
He could see the problems ahead in dealing with the tools,,time I could better spend learning all the other aspects of 'the trade' as he called it.
The flat/wriggle tool in the pics is a 1/8" Cobalt tool bit. I buy the 2 1/2" lengths,,they work out well for both hammer and graver tools. A couple of packets will last a full timer a long time.
If you get in to carbide bits, you'll need diamond sharpening hones and the like. Not really necessary for most work as seen here unless you enjoy cutting through case hardened surfaces. That's not a good way to do the work anyway though some think it is acceptable.
The handle on the one pictured is a piece of dowel and the furrel is a piece of cartridge brass.
Take care to shape the handle, it makes a lot of difference in your cutting as fatigue and muscle strain set in from an uncomfortable grip.
If you're doing more than just wriggle cutting an initial or two, perhaps your name on a bbl,,,long days at the engravers bench can take a toll on the body. Take every advantage you can.
Pretty fancy graver for me. Doesn't take long to make up. At todays price the bits are about $3. No heat treating or grinding necessary except for what angle you want.
Been using that one for a long time.
Just some thoughts,,
Good luck to you in your pursuits.