All well-produced human voices have a degree of quaver known as vibrato. Some are given a wide, cello-like vibrato, others a mere wiggle; but most singers agree that there is a natural vibrato which is sometimes discovered at an early age, occasionally in midlife (like my own), and which increases with age. Gossipy sopranos might describe the wobble in an aging peer: "she's got a tremolo you could drive a truck through!"
Fashions and styles ask for varying degrees of vibrato. Folk prefers a tight little tremble, suitable for urgency of lyric and clarity of text. Grand opera, which needs huge, free voices to fill auditoriums, asks for large vibrato, as does the inner-city Gospel choir with Mahalia Jackson its role model. Big Band era singers started with a straight tone and let it open up into a vibrato, like a Leslie B-3 when you turn the juice on the rotor. St. Olaf's in Minnesota demands a crystal-clear, vibrato-free sound to emulate the famous boys' choirs of Lutheran history, and let the lines of polyphony be heard distinctly. Each requires both a certain native vibrato, and/or the willingness of a singer to alter their vibrato, or be forbidden to participate.
It seems, in my limited "neck of the woods", that the free vibrato has become anathema to the contemporary music world, somehow having become associated with "old things" and (shudder) "old people." I am a bit tired of hearing of singers with vibrato being turned out of the worship team or being put on a list of acceptable leaders for Wednesday Night adult Bible Study (you don't want to expose young people to vibrato -- they'll surely leave the church).
I'm looking for input from a wider variety of churches than those in my neighborhood. Do you allow people with vibrato (I mean a good vib w/o wobble, and good pitch) to lead worship or sing on the team? Have you encountered marginalization on account of vibrato, either presence or lack of? Or do people just like what they like, and that is all there is to it?
My wife, who has had a pleasing soloist's natural vibrato since the age of six, tells me that all people have some measure of vibrato. Your description of your voice is about normal - and since you know where its limits are, you're better off than most! My baritone is about the same range as yours, and straight as a stick most of my life; I've picked up a little pleasant waver through practicing, after a vocal operation; it caused me to intentionally relax before singing. Laurel says relaxation is the key, just as flute players tell you it's the key to getting tone out of that mysterious implement (which I can do for one note, but when I change the fingering, it's gone).
Perhaps the term "correctly" is the rub. We straights start trying, and the instant the vibrato-person sees our eyebrows move or anything tighten AHA! YOU'RE NERVOUS. RELAX RELAX RELAX!!!!! And of course, we become vibratoless basket cases. Let's just enjoy our bellowing, for it be to the Lord, and His ears love the praise.