If responder to a convenient minor 1 bid has 5 points or less I have been told the correct bid is 1 notrump. Is a convenient minor 1 bid a demand bid?
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The Convenient Minor as a forcing bid is a prohibited agreement in North America (ACBL Jurisdiction). To get caught regularly responding to it with less than 5 points will result in severe disciplinary action by the ACBL.
If you are not playing in North America, you must refer to the agreed conventions and agreements as per your national authority on the rules. In most countries such unusual agreements are typically prohibited except in extended team play competition, just as in North America.
I would never respond 1NT to 1m with fewer than 5 HCP. with some partners, I only respond at the 1 level with sound values. With others, I'll respond 1 of a suit with a weak hand whenever I lack 4-card support for their clubs or 3-card support for their diamonds. It's simply a matter of partnership preference; I don't think one way or the other is markedly superior; both leave you in trouble some of the time.
One point you and your partner should consider is whether you open strong 2-suiters with 2C or prefer to open 1 of your stringer suit and then jump-shift or reverse. If you sometimes open hands that nearly have game in hand 1m, you should respond with weak hands.
The permitted "convenient minor" forcing sequence is 1 club, pass, 1 diamond. In this case, the opener must have at least 10 high card points (hcp), if responder may have nothing.
A strong "minor" forcing sequence is 2 clubs, pass, 2 diamonds, where the opener has at least 22-24, and the responder might have "nothing.
Besides being prohibited, the sequence 1 diamond, pass, 1NT (with fewer than 5 hcp), doesn't work well because you skip two bids, hearts and spades, depriving both your team and your opponents of them. The ACBL frowns on sequences whose primary impact is to disrupt your opponents' bidding without helping your own cause (that is, sequences that hurt you some, and hurt your opponents more).
In the club-bid, diamond-response sequences, the response didn't take up any additional bidding space. Because it takes up bidding space, the 1NT response to 1 diamond must be "natural" (typically 6-9 hcp, although 5 will be permitted), and reasonably balanced.