[Trick13]

1♥ (P) 4♥ (4♠)
X

When 1♥ is limited, the double shows a strong desire to compete to the 5-level but is giving responder an opportunity to defend if the 4♥ raise was based on strength.

Should X be the same in the following auctions:

1♥ (1♠) 4♥ (4♠)
X

1♥ (2♠*) 4♥ (4♠) * weak
X

1♥ (3♠) 4♥ (4♠)
X

1NT* (2♠) 4♥ (4♠) * weak, possibly off-shape
X

Is an ambiguous raise to 4♥, given the seeming inevitability of 4♠ these days, a good idea?

[sfi]

Trick13, on 2018-November-05, 16:46, said:

1♥ (P) 4♥ (4♠)
X

When 1♥ is limited, the double shows a strong desire to compete to the 5-level but is giving responder an opportunity to defend if the 4♥ raise was based on strength.

Should X be the same in the following auctions:
(auctions removed)

In my experience, yes. There's no real reason to treat any of them differently than the first one.

Quote

Is an ambiguous raise to 4♥, given the seeming inevitability of 4♠ these days, a good idea?

You should certainly be prepared for the decision over 4S and, if it gives you an uncomfortable choice, consider other approaches to the hand. Partner will bid 5H much more rarely with the style you are playing, so you will have more information most of the time before you have to make that decision. I often raise to 4H immediately if I am comfortable defending 4S (maybe doubled), am comfortable bidding 5H but don't think they have a slam (so they might go wrong at the five-level) or if I want to give partner a chance to suggest 5H.

[apollo1201]

For the competitive sequences, in the first 2 ones, where bidding space isn’t cramped, you have ways to show a good GF raise based on strength (« the hand belongs to us, p ») rather than a competitive « LoTT » raise. Should this be taken into consideration (forcing passes, X rather thinking not to go to the 5-level, etc.)?

[sfi]

apollo1201, on 2018-November-06, 01:32, said:

For the competitive sequences, in the first 2 ones, where bidding space isn’t cramped, you have ways to show a good GF raise based on strength (« the hand belongs to us, p ») rather than a competitive « LoTT » raise. Should this be taken into consideration (forcing passes, X rather thinking not to go to the 5-level, etc.)?

One of the big advantages of playing a system where 1M is limited (Precision and Polish Club, to mention two) is being able to jump to 4M on a wide range of hands. That makes it harder for the opponents to judge accurately at that level. If you deny yourself that option, your opponents will know when it is right to compete. Playing such a system, you should only use cue raises and the like when you don't know where the hand is heading and want to involve partner.

The agreement described in the opening post is used to give opener a way to compete to the five level while not getting in the way of a partner who jumps to 4H with the intent of penalising a reflex 4S bid.

[apollo1201]

sfi, on 2018-November-06, 03:24, said:

The agreement described in the opening post is used to give opener a way to compete to the five level while not getting in the way of a partner who jumps to 4H with the intent of penalising a reflex 4S bid.

Even with a 11-15 or so partner you must be pretty well aligned on who does what (I guess with a strong hand you reopen with X when partner passes?), especially when after an intervention you expect the sequence to turn really competitive and quickly reach high levels.
Admittedly, I played a précision-like system too long ago with 1 partner only, for an event (and loved the unopposed 1-4 sequence) but am probably keeping reflexes of more wide-ranging opening systems!