Answering the next question is a bit trickier. You said "nets (profit) me $5k/month" which can be interpreted in a number of ways. So I'll answer with the two most common interpretations.
If you're SaaS business can pay you and anyone else needed to run it, a salary and some benefits, and at the end of the year, after accounting for depreciation of assets, expenses, and taxes is $5K in the 'black' (so the LLC, S-Corp, what have you, could 'bank' that $5K for future expansion) then no it isn't a start up any more. It may be a boutique business but its a going concern.
If you run a SaaS business with no employees, and use the revenue to pay your living expenses, have no benefits and you happen to end up with $5K unspent at the end of the year, its more of a consultancy than a startup.
Good examples in an adjacent field are accounting companies versus accounting consultants. My mother-in-law ran her own little tax accounting business for years. Not a startup (it was self sustaining) but not large either.
The term is not well defined, and basically meaningless; but I think a definition that would resonate more closely with most people is "a company whose meta-strategy is to expand like crazy, whose strategy is flexible, and who is not yet an industry pillar".
Or grow a sustainable business from something small to something huge that produces a lot of cash along the way while your enterprise value increases ahead of a large exit (or die trying).