I want to use the name Leonardo da Vinci name for a presentation, but I I can't fit his whole name and would only like to use "da Vinci" My question is...which is more proper, using "Da Vinci" with a capital "D" or "da Vinci" with a lowercase "d"?
His whole name is just "Leonardo". The 'da Vinci' part means "from Vinci". Typically in old days people had just one name and they were further identified by their profession, birthplace or who their father was. Typically the 'da' part is not capitalized because it is just a preposition.
Furthermore contemporary people would not call him just "da Vinci", they should always call him "Leonardo" or "Leonardo da Vinci" if just Leonardo isn't identifying enough. If even that isn't unique enough (like if you're in Vinci, there's probably multiple Leonardos), then he is the son of gentleman Piero, so "Leonardo di ser Piero" or "Leonardo di ser Piero da Vinci". But never just "da Vinci".
Of course, in modern days we often apply the modern styling of names to him and treat "da Vinci" as if it was his surname. But still, when used, whether alone or with his name, it should not be capitalized, unless of course it happens at the start of the sentence.
This isn't a problem in just Italian but many languages use un-capitalized words that are the equivelent of "from" or "of" as part of their name. In all cases, it should be followed by that language's customs, not Engilsh as names are never "translated" in a shared writing system (there are rules for Transliteration between two alphabets). Typically, rendering the proper last name is to follow the way the person presents it) So that Leonardo da Vinchi would be "da Vinchi" if refering to him by his last name, to distinguish him from "DiCaprio" or "the Ninja Turtle" (the actor is an example where the two word last name was changed to a single name with two capitilizations, probably as part of immigration officials not knowing the proper spelling... which is common in non-English last names). Other famous people include the former U.S. President with the Dutch last name "van Buren" and the German-American Rocket Scientist "von Braum".
Though this format isn't always used with all languages so you will have to check. A person with a Portugese name like "Leonardo Santa Cruz de Silvia" would be addressed as "Mr. Silvia" if following proper language customs (and just so we're clear, most names in Spanish and Portuguese, the middle name "Santa Cruz" is usually the mother's family name, not a given middle name as it would be in English).
It's best to identify the root language of the name, and follow the customs unless the individual gives a preference.
All of the answers seemed from the point of view of the Italians at the time of Leonardo da Vinci's life.
The OP is writing a presentation in the current day. In the current day if I saw a slide with "da Vinci" on it, I would fill in Leonardo, regardless of the capitalization.
Though, if I couldn't fit it all in one line, I'd use two lines so I could get his first name there. I wouldn't care so much about the grammar, but I feel he deserves a first name. I admire the guy that much.