How could I create a string pattern in Lua to find encapsulated text similar to markdown?

I'm still getting used to the concept of string patterns, and I've run into an issue regarding them. I'm trying to create a simple program that searches a string of text, for certain characters encapsulated in whatever the brackets may be. Here's an example:

local str = "Hello <<world>>, my <<name>> is <<John>>"

-- Match patterns with << ... >>
for noun in str:gmatch("<<.->>") do
    print(noun)
end

This program will search through the string, matching everything that starts with << and ends with >>, and everything in between. Good, that's what I want. However, let's say I wanted a different pattern that only got text between one of those tags instead of two (< and > instead of << and >>). This is where I run into a problem:

-- Allow easy customization control over brackets
local matchNouns = {"<<", ">>"}
local matchOther = {"<", ">"}

local str = "<Hello> <<world>>, <my> <<name>> <is> <<John>>"

local function printOtherMatches(str)
    -- Get opening and closing brackets
    local open, close = unpack(matchOther)

    -- Concatenate opening and closing brackets with
    -- pattern for finding all characters in between them
    for other in str:gmatch(open .. ".-" .. close) do
        print(other)
    end
end

printOtherMatches(str)

The program above will print everything between < and > (the matchOther elements), however it also prints text captured with << and >> as well. I only want the iterator to return patterns that explicitly match the opening and closing tags. So the output from above should print:

<<Hello>>

<<my>>

<<is>>

Instead of:

<<Hello>>

<world>

<<my>>

<name>

<<is>>

<<John>>

Basically, just like with markdown how you can use * and ** for different formats, I'd like to create a string pattern for that in Lua. This was my attempt of emulating that kind of pattern sequence. If anyone has any ideas, or insight on how I could achieve this, I'd really appreciate it!