Learn English – Appointment vs. meeting (vs. date)

Neither of OP's alternatives are acceptable compressed forms of what would normally be referred to as date of starting work in the full form of this compound noun. The most common short form is...

work start date

Note that the base noun here is date - modified by another noun start, which is itself modified by the noun work. Most compound nouns involve only two nouns anyway, and if there are more than two they usually all modify the same base noun, but the only grammar involved here is that when a noun is being used "adjectivally" it follows the normal rule for adjectives - it comes immediately before the noun it modifies (even if that modified noun is then used adjectivally to modify another noun that follows).

You might do a meeting if you're in Los Angeles

P1: Let's do a meeting after lunch
P2: I don't do meetings

if you attended a meeting

P1: Were you able to make that meeting?

P2: No, I was running late and didn't make the meeting
P2: Yes, I made the meeting

meaning

...didn't make it to the meeting
I made it to the meeting

The phrase you are looking for is

have a meeting

We would like to have a meeting to discuss the details
We would like to meet to discuss the details

Both imply a face-to-face meeting.