In ancient Athens, direct taxation of the citizenry seems to be strongly associated with tyranny, so while there were taxes on imported goods and resident foreigners (metics) paid a regular tax, citizens were expected to pay taxes voluntarily.

There was something like an agricultural tax in that the "first fruits" of every harvest were expected to be given to state-sponsored cults. It was tied to religion as well as civic values, so it doesn't appear to have been enforced by tax collectors so much as the community in general. Also the very wealthy were expected to pay liturgies which included things like outfitting a trireme or sponsoring a tragedy/games. If someone was nominated to pay a liturgy but they thought there was someone else who was wealthier and who hadn't paid in a longer period of time, they could challenge them and then the person had to either 1) pay the liturgy or 2) swap assets with the person who challenged them and then pay the liturgy with the other person's money.

Of course, a lot of this reflects the practice during the "golden age of Athens" when the polis collected money from other poleis and also from the silver mines at Laurium (worked by slaves), so taxes were not the only source of civic revenue. Their voluntary taxation system didn't seem to hold out so well in the age of empires that followed.

This thread, and this comment are more or less what you're looking for as far as I can tell.

But as urbanization increased, how did census takers account for city dwellers who owned little land but more fluid property?

Generally speaking this was a non-issue for the Roman state. Tax revenue was collected on a provincial scale, with local promagistracies or procuratorships (in the Principate) bearing responsibility for facilitating tax collection across the entire province (the precise means depending on the period), without necessarily worrying about actual distributions of wealth, which could cause problems sometimes. Publican contracts were sold in the late Republic by auction, with different publican companies bidding different amounts of money to the state. The winning bidders paid their bids up-front, meaning that whatever money they collected in their alloted province was their own to recoup their losses. The major problem, of course, was that publicans generally extracted more money than they had bid, to male a profit, leading to hardship in the provinces--apart from this provincial promagistrates in the late Republic generally had rights to extract special grain taxes or provincial aid for the maintenance of their armies. Unscrupulous governors could and frequently did extort additional money and grain, selling the excess and pocketing the profits. Some provinces had more fixed tax procedures, and the Principate brought much more centralized collection, but even in these provinces (such as Sicily, where Verres terrorized the local population despite relatively set tax standards) in the late Republic (and even the Principate) the enormous powers of provincial promagistrates and the general lack of oversight over publicans led often to rather ad hoc taxes and collection